Generalization of prime ideals in $M_n(N)$-group $N^{n}$

Abstract

The notion of a matrix nearring over an arbitrary nearring was introduced by (Meldrum and Walt Arch. Math. 47(4): 312–319, 1986). In this paper, we define the notions such as weakly $\tau$-prime $(\tau =0,c,3,e)$ ideals of an N-group G,  which are the generalization of the classes of $\tau$-prime ideals of G, and provide suitable examples to distinguish between the two classes. We extend the concept to obtain the one-one correspondence between weakly $\tau$-prime ideals $(\tau =0,c,3,e)$ of N-group (over itself) and those of $M_n(N)$-group $N^{n}$, where $M_n(N)$ is the matrix nearring over the nearring N. Further, we prove the correspondence between weakly 2-absorbing ideals of these classes.

1 Introduction

Nearrings are generalized rings where the addition need not be abelian and only one distributive property is allowed. Rings can be viewed as algebraic systems of ‘linear’ functions on groups, while nearrings describe the general non-linear case [2]. Matrix nearrings over arbitrary nearrings were introduced by Meldrum & Van der Walt [1], wherein the correspondence between the two-sided ideals in nearring N and those of matrix nearring $M_n(N)$ were obtained. We denote N for a zero-symmetric (right) nearring with 1, and G for an N-group (denoted as, ). Considerable developments in matrix nearrings over arbitrary nearrings were due to Meyer [3], Booth and Groenewald [4]. Meyer [3] proved that for a left ideal L of N, $L^n$ is its corresponding ideal in $M_n(N)$-module $N^n$. Van der Walt [5] studied the relationship between primitive modules over a nearring N and those of the matrix nearring $M_n(N)$. Indeed, when G is a locally monogenic N-group, then the action of $G^{n}$ over $M_n(N)$ was defined. However, for an N-group (over itself), $N^n$ becomes an $M_n(N)$-group. The motivation to study the interrelations between the N-group over N and the $M_n(N)$-group $N^n$ is an analogy to the concept of Morita equivalence. Two unital rings are Morita equivalent if they have equivalent categories of left modules. If A is a left R-module, then $A^{n}$, $n\in \mathbb {N}$ is a $M_n(R)$-module in a canonical way. The correspondence $A\rightarrow A^{n}$ preserves homomorphisms and conversely every $M_n(R)$-module can be obtained in this way. In this paper, we introduce the concepts weakly $\tau$-prime $(\tau =0,c,3,e)$ ideals of an N-group G and obtain the one-one correspondence between weakly $\tau$-prime ideals of N-group (over itself) and those of $M_n(N)$-group $N^{n}$. As a generalization of a c-prime and a 3-prime ideal of N-group G, we define the notions weakly (c, 2)-absorbing ideal and weakly (3, 2)-absorbing ideal, respectively, and provide suitable examples. Further, we prove the one-one correspondence between (i, 2), $i\in \{c,3\}$-absorbing ideals of N (over N) and those of $M_n(N)$-group $N^{n}$. Bhavanari et.al [6] proved the correspondence between the prime left ideals of N and that of $M_n(N)$. Juglal et.al [7] studied different prime N-ideals and prime relations between generalized matrix nearring and multiplication modules over a nearring. Further, Juglal and Groenewald [8] studied the class of strongly prime nearring modules and shown that it forms a $\tau$-special class. In Bhavanari and Kuncham [9], the relation between the ideals of the N-group N and the ideals of $M_n(N)$-group $N^n$ has been studied. Badawi and Darani [10] studied weakly 2-absorbing ideals as a generalization of prime ideals in commutative rings.

For standard notations and definitions in nearrings, we refer to Pilz [11], Bhavanari and Kuncham [2].

We denote $H\trianglelefteq _{N} G$ for an ideal H of G (if H is proper, we use $H\triangleleft _{N} G$).

From [2], for any $u\in G$, the ideal generated by u is denoted by $\langle u \rangle$ and defined as, $\langle u \rangle =\displaystyle \bigcup _{i=1}^{\infty }S_{i+1}$, where $S_{i+1}=S_{i}^{*}\cup S_{i}^{0}\cup S_{i}^{+}$ with $S_{0}=\{u\}$, and $S_{i}^{*}=\{g+y-g: g\in G, y\in S_{i}\},$ $S_{i}^{0}=\{p-q: p,q\in S_{i}\}\cup \{p+q:p,q\in S_{i}\},$ $S_{i}^{+}=\{n(g+a)-ng: n\in N, g\in G, a\in S_{i}\}.$

Definition 1.1

[7, 12] Let $P\vartriangleleft _{N}G$ such that $NG\nsubseteq P$. Then

• 1. P is prime (or 0-prime) if for every ideal A of N and every ideal B of G, $AB\subseteq P$, then $AG\subseteq P$ or $B\subseteq P$,

• 2. P is 3-prime if for $n\in N$ and $g\in G$, $nNg\subseteq P$, then $nG\subseteq P$ or $g\in P$.

• 3. P is completely prime (denoted as, c-prime) if for $n\in N$ and $g\in G$, $ng\in P$, then $nG\subseteq P$ or $g\in P$.

• 4. P is equiprime (denoted as, e-prime) if $a\in N$ and $g_1,g_2\in G$, $ang_1-ang_2\in P$, for all $n\in N$, then $aG\subseteq P$ or $g_1-g_2\in P$.

2 Weakly prime ideals of N-group

We define weakly $(\tau =0,c,3,e)$-prime ideal of an N-group G and provide examples to distinguish between the classes weakly $\tau$-prime ideals and $\tau$-prime ideals of G.

Definition 2.1

An ideal P of G with $NG\nsubseteq P$ is called a weakly 0-prime ideal (or simply weakly prime ideal) of G, if for any ideal A of N and ideal B of G, $(0)\ne AB\subseteq P$, then $AG\subseteq P$ or $B\subseteq P$.

Proposition 2.2

The following conditions are equivalent for an ideal P of G.

• 1. P is a weakly prime ideal of G.

• 2. If $x\in N$, $y\in G$, $\left( 0 \right) \ne \langle x \rangle \langle y \rangle \subseteq P$, then $xG \subseteq P$ or $y \in P.$

• 3. If $x\in N$, $y\in G$, $\left( 0 \right) \ne x \langle y \rangle \subseteq P$, then $xG \subseteq P$ or $y \in P.$

• 4. If $x\in N$, $y\in G$, $\left( 0 \right) \ne \langle x \rangle \left( P+ \langle y \rangle \right) \subseteq P$, then $xG \subseteq P$ or $y \in P.$

• 5. If A is an ideal of N, B an ideal of G and $\left( 0 \right) \ne A\left( P+B\right) \subseteq P$, then $AG \subseteq P$ or $B \subseteq P.$

Proof

(1)$\Rightarrow$(2): Suppose that P is a weakly prime ideal of G.

Let $\left( 0 \right) \ne \langle x \rangle \langle y \rangle \subseteq P.$ Then by (1), $\langle x \rangle G \subseteq P$ or $\langle y \rangle \subseteq P$. Now $xG\subseteq \langle x \rangle G \subseteq P$ or $y\in \langle y \rangle \subseteq P$, implies $xG \subseteq P$ or $y \in P.$

(2)$\Rightarrow$(3) is obvious.

(3)$\Rightarrow$(4): Suppose that $\left( 0 \right) \ne \langle x \rangle \left( P+ \langle y \rangle \right) \subseteq P.$ Then there exists $(0) \ne x\langle y\rangle \subseteq \langle x \rangle \left( P+ \langle y \rangle \right) \subseteq P$. This implies $\left( 0\right) \ne x\langle y\rangle \subseteq P.$ Now by (3), $xG \subseteq P$ or $y \in P.$

(4)$\Rightarrow$(5): Suppose that $\left( 0 \right) \ne A\left( P+B\right) \subseteq P.$ In a contrary way, suppose that $AG \nsubseteq P$ or $B \nsubseteq P.$ Then there exist $xg\in AG\backslash P$ and $y \in B\backslash P ,$ for some $x\in A$, $g\in G$, which implies $xg \notin P$ or $y \notin P.$ Then by (4), $\left( 0 \right) \ne \langle x \rangle \left( P+ \langle y \rangle \right) \nsubseteq P.$

But $\langle x \rangle \left( P+ \langle y \rangle \right) \subseteq A(P+B) \subseteq P,$ a contradiction. Therefore, $AG \subseteq P$ or $B \subseteq P.$

(5)$\Rightarrow$(1): Let A be an ideal of N and B an ideal of G such that $\left( 0\right) \ne AB \subseteq P.$ In a contrary way, $AG \nsubseteq P$ and $B \nsubseteq P.$ Then by (5), $\left( 0 \right) \ne A\left( P+B\right) \nsubseteq P,$ which implies there exist $x \in A$ and $y\in B$ such that $0 \ne xy \in A\left( P+B\right) \nsubseteq P$. This implies, $0 \ne xy \in P,$ a contradiction. Therefore, $AG \subseteq P$ or $B \subseteq P.$ $\square$

Definition 2.3

An ideal I of G with $NG\nsubseteq I$, is called

• 1. weakly c-prime if $n\in N$, $g\in G$ and $0\ne ng\in I$, then $nG\subseteq I$ or $g\in I$.

• 2. weakly 3-prime if $n\in N$, $g\in G$ and $(0)\ne nNg\subseteq I$, then $nG\subseteq I$ or $g\in I$.

• 3. weakly e-prime if $a\in N$, $g_1,g_2\in G$ and $0\ne ang_1-ang_2\in I$, for all $n\in N$, then $aG\subseteq I$ or $g_1-g_2\in I$.

Proposition 2.4

Let be an ideal of G such that $NG\nsubseteq P$. Then P is

• 1. weakly c-prime implies weakly 3-prime;

• 2. weakly 3-prime implies weakly prime;

• 3. weakly e-prime implies weakly 3-prime.

Proof

(1) Suppose P is a weakly c-prime ideal. Let $a\in N$ and $g\in G$ such that $(0)\ne aNg\subseteq P$. To show $aG\subseteq P$ or $g\in P$. If $g\in P$, it is clear.

Case-(i) Let $g\notin P$. If $(0)=Ng$, then since $1\in N$, $0=g$, a contradiction. Now assume that, $(0)\ne Ng\subseteq P$. Now, $0\ne ag\in P$, for all $a\in N$. Since P is weakly c-prime and $g\notin P$, we get $aG\subseteq P$.

Case-(ii) Let $g\notin P$ and $(0)\ne Ng\nsubseteq P$. Then, there exists $n\in N$ such that $0\ne ng\in P$, but $0\ne ang\in P$. Since P is weakly c-prime and $g\notin P$, we get $aG\subseteq P$. Therefore, P is weakly 3-prime.

(2) Suppose P is weakly 3-prime. Let A be an ideal of N and B be an ideal of G such that $(0)\ne AB\subseteq P$. To show $AG\subseteq P$ or $B\subseteq P$. If $B\subseteq P$, it is clear. If $B\nsubseteq P$, there exists $x\in B\setminus P$, and since $1\in N$, for every $a\in A$, we have $(0)\ne aNx\subseteq ANB\subseteq AB$ (by Pilz [11], Proposition 1.34) $\subseteq P$. Since P is weakly 3-prime and $x\notin P$, it follows that $aG\subseteq P$, for all $a\in A$. Therefore, $AG\subseteq P$, shows that P is weakly prime.

(3) Suppose P is weakly e-prime. Let $a\in N$ and $g\in G$ such that $(0)\ne aNg\subseteq P$. To show $aG\subseteq P$ or $g\in P$. If $g\in P$, it is clear. So, let $g\notin P$. Since $(0)\ne aNg\subseteq P$, $0\ne ang\in P$, for all $n\in N$. Now $0\ne ang=ang-an0_{G}\in P$, as N is zero-symmetric. Since P is weakly e-prime and $g=g-0_{G}\notin P$, we get $aG\subseteq P$. Therefore, P is weakly 3-prime. $\square$

Example 2.5

• 1. Consider $N=(\mathbb {Z},+,\cdot )$ and $G=((\mathbb {Z}_4\times \mathbb {Z}_2),+)$. Then G is an N-group. The ideal $\langle (0,1)\rangle$ is weakly prime, but not prime, as $2\mathbb {Z}\cdot \langle (2,1)\rangle =\langle (0,0)\rangle \subseteq \langle (0,1)\rangle$, but $\langle (2,1)\rangle \nsubseteq \langle (0,1)\rangle$ and $2\mathbb {Z}\cdot G\nsubseteq \langle (0,1)\rangle$.

• 2. Let $p\ne q$, be distinct prime. Consider $N=(\mathbb {Z},+,\cdot )$ and $G=(\mathbb {Z}_{pq},+_{pq})$. Then G is an N-group. Here, (0) is weakly prime ideal but not prime.

• 3. Let $N=\{0,a,b,c\}$ ([11], Table E(8)), $G=N$. Then G is an N-group over itself.

  $+$	0	a	b	c
  0	0	a	b	c
  a	a	0	c	b
  b	b	c	0	a
  c	c	b	a	0

  $\cdot$	0	a	b	c
  0	0	0	0	0
  a	0	0	0	a
  b	0	a	b	b
  c	0	a	b	c

  The ideals of G are $I_{1}=\lbrace 0,a \rbrace , I_{2}=\lbrace 0,b\rbrace , I_{3}=\lbrace 0,c \rbrace .$ An easy observation yield, $I_{1}I_{2}=\lbrace 0\rbrace ,~ I_{2}I_{1}=I_{1},~ I_{2}I_{3}=I_{2},~ I_{3}I_{2}=I_{2},~ I_{3}I_{1}=I_{1},~ I_{1}I_{3}=I_{1}.$

  • (a) $I_1$ is weakly prime as well as prime.

  • (b) $I_{2}$ is neither weakly prime nor prime, since $(0)\ne I_{2}I_{3}=I_{2}$, but $I_{2}G \nsubseteq I_{2}, I_{3}\nsubseteq I_{2}.$

  • (c) $I_{3}$ is weakly prime but not prime, since $I_{1}I_{2}=\lbrace 0 \rbrace \subseteq I_{3},$ and $I_{1}G \nsubseteq I_{3}, I_{2}\nsubseteq I_{3}.$

  • (d) $I_2$ is weakly c-prime, but not c-prime, since $a.b=0\in I_{3}$, and $aG\nsubseteq I_{3}$, $b\notin I_{3}$.

  • (e) $I_3$ is weakly 3-prime, but not 3-prime, since $aNb=0\in I_{3}$, and $aG\nsubseteq I_{3}$, $b\notin I_{3}$.

  • (f) $I_2$ is weakly 3-prime, but not 3-prime, since $aN=0\in I_{3}$, and $aG\nsubseteq I_{3}$, $a\notin I_{3}$.

  • (g) $I_2$ is weakly e-prime, but not e-prime, since $0=ana-anb\in I_{2}$, and $aG\nsubseteq I_{2}$, $a-b\notin I_{2}$.

• 4. Let $(N,+)$ be any group with at least 3 distinct non-zero elements, say $a,g_1,g_2$. Define multiplication on N by

  $$\begin{aligned} ab= \left\{ \begin{array}{ll} a &{} {\text{ if }} b\ne 0,\\ 0 &{} {\text{ if }} b=0. \end{array} \right. \end{aligned}$$

  Then $(N,+,\cdot )$ is a zero-symmetric nearring. Here, $\{0\}$ is weakly e-prime ideal but not e-prime, since $0\notin \{a,g_1,g_2\}\subseteq N$ such that $g_1\ne g_2$ and $aN\nsubseteq \{0\}$. However, for each element $n\in N$,

  $$\begin{aligned} ang_1= \left\{ \begin{array}{ll} an &{} \text{ if } g_1\ne 0,\\ 0 &{} \text{ if } g_1=0. \end{array} \right. ,~ ang_2= \left\{ \begin{array}{ll} an &{} \text{ if } g_2\ne 0,\\ 0 &{} \text{ if } g_2=0.\\ \end{array} \right. \end{aligned}$$

  Thus, $ang_1-ang_2=0$.

3 Weakly prime ideals of $M_n(N)$-group $N^n$

Now we give some basic definitions about the matrix nearring from [1]. For a zero-symmetric right nearring N with identity 1,  let $N^{n}$ will denote the direct sum of n copies of $(N,+)$. The elements of $N^{n}$ are column vectors and written as $\left( r_1,\ldots ,r_n\right)$. The symbols $i_{j}$ and $\pi _{j}$ respectively, denote the $i^{th}$ coordinate injective and $j^{th}$ coordinate projective maps.

For an element $a\in N$, $i_{i}(a)=(0,\ldots ,\underbrace{a}_{i^{th}},\ldots ,0)$, and $\pi _{j}(a_1,\ldots ,a_n)=a_{j}$, for any $(a_1,\ldots ,a_n)\in N^n$. The nearring of $n\times n$ matrices over N, denoted by $M_n(N)$, is defined to be the subnearring of $M(N^{n})$, generated by the set of functions {$f_{ij}^{r}:N^{n}\rightarrow N^{n}~|~r\in N,1\le i,j\le n$} where $f_{ij}^{r}\left( r_1,\ldots ,r_n\right) :=\left( s_1,s_2,\ldots ,s_n\right)$ with $s_i=rr_j$ and $s_k=0$ if $k\ne i$. Clearly, $f_{ij}^{r}=i_{i}f^{r}\pi _{j}$, where $f^{r}:N\rightarrow N$, $r\in N$ by $f^{r}(x)=rx$, for all $x\in N$. If N happens to be a ring, then $f_{ij}^{r}$ corresponds to the $n\times n$-matrix with r in position (i, j) and zeros elsewhere. Following the notation from ([9], Notation 1.1), for any ideal $\mathcal {I}$ of $M_n(N)$-group $N^n$, we write

$$\begin{aligned} \mathcal {I}_{**}=\{a\in N:a=\pi _{j}A,~\text {for some}~A\in \mathcal {I},1\le j\le n\}, \end{aligned}$$

where $\pi _{j}$ is the $j^\text {th}$ projection map from $N^{n}$ to N.

Throughout, we denote $M_n(N)$ for a matrix nearring, $N^n$ for an $M_n(N)$-group $N^n$; 0, $\bar{\mathbf{0 }}$ and $\bar{0}$ respectively denote the additive identities of N, $M_n(N)$ and $N^n$. As usual, $I^{n}$ denotes the direct sum of n copies of an ideal I of N. Furthermore, we refer to Meldrum & Van der Walt [1] for detailed literature on matrix nearrings. Now, we introduce the notions weakly $\tau$-prime $(\tau =0,c,3,e)$ ideals of $M_n(N)$-group $N^n$ and prove the correspondence between the weakly $\tau$-prime ideals $(\tau =0,c,3,e)$ of N-group (over itself) and those of $M_n(N)$-group $N^{n}$.

The following Lemma 3.1 and Lemma 3.3 from [9] are useful, and so, for completeness we brief the proofs.

Lemma 3.1

For any $t\in N$, let $i_{i}(t)=\left( 0,\ldots ,t,\ldots ,0\right)$ with t in the $i^{th}$ position. Then $\langle i_{i}(t)\rangle =\langle (t,0,\ldots ,0)\rangle$.

Proof

Suppose t is an element of N. Then $(t,0,\ldots ,0)\in N^{n}$. Now $(t,0,\ldots ,0)$

$=f_{1i}^{1}(i_{i}(t))\in \langle i_{i}(t)\rangle$. Therefore, $\langle (t,0,\ldots ,0)\rangle \subseteq \langle i_{i}(t)\rangle$. On the other hand, $i_{i}(t)=f_{1i}^{1}(t,0,\ldots ,0)\in \langle (t,0,\ldots ,0)\rangle$. Therefore, $\langle i_{i}(t)\rangle \subseteq \langle (t,0,\ldots ,0)\rangle$. $\square$

Lemma 3.2

If s, t are elements of N and $s\in \langle t\rangle$, then

$(s,0,\ldots ,0)\in \langle (t,0,\ldots ,0)\rangle$.

Lemma 3.3

If t is an element of N, then $\langle t\rangle ^{n}=\langle (t,0,\ldots ,0)\rangle$.

Proof

Since $(t,0,\ldots ,0)\in \langle t\rangle ^{n}$, we have $\langle (t,0,\ldots ,0)\rangle \subseteq \langle t\rangle ^{n}$. Take $(s_1,s_2,\ldots ,s_n)\in \langle t\rangle ^{n}$. This implies $s_i\in \langle t\rangle ^{n}$, for $1\le i\le n$. By Lemma 3.2, $(s_i,0,\ldots , 0)\in \langle (t,0,\ldots ,0)\rangle$ for all i. Now, from ([11], Proposition 1.34), it follows that

$$\begin{aligned} (s_1,s_2,\ldots ,s_n)&=f_{11}^{1}(s_1,0,\ldots ,0)+f_{21}^{1}(s_2,0,\ldots ,0)\\&~~~~~+\ldots +f_{n1}^{1}(s_n,0,\ldots ,0)\\&\in \langle (t,0,\ldots ,0)\rangle . \end{aligned}$$

$\square$

Definition 3.4

An ideal $\mathcal {I}$ of $N^{n}$ with $M_n(N)N^{n}\nsubseteq \mathcal {I}$ is said to be a weakly prime ideal if for any ideal $\mathcal {A}$ of $M_n(N)$, and an ideal B of $N^{n}$ with $(\bar{0})\nsubseteq \mathcal {A}B\subseteq \mathcal {I}$, then either $\mathcal {A}N^{n}\subseteq \mathcal {I}$ or $B\subseteq \mathcal {I}$.

Proposition 3.5

If I is a weakly prime ideal in $_{N}^{}{N}$, then $I^{n}$ is a weakly prime ideal in $M_{n}(N)$-group $N^{n}.$

Proof

Let I be weakly prime in $_{N}^{}{N}$. By ([1], Proposition 4.1), $I^{n}$ is an ideal of $N^{n}$. To prove, $I^{n}$ is weakly prime in $N^{n}$, let $\mathcal {A}$ an ideal of $M_n(N)$ and K be an ideal of $N^{n}$ such that $\mathcal {A}N^{n} \nsubseteq I^{n} ~\text {and}~K\nsubseteq I^{n}.$ By ([9], Lemma 1.5), $K=B^{n}$ for some ideal B of $_{N}^{}{N}$. Then there exist $A\in \mathcal {A}$, $\rho \in N^{n}$ and $(b_1,b_2,\ldots ,b_n)\in B^{n}$ such that $A\rho \notin I^{n}$ and $(b_1,b_2,\ldots ,b_n)\notin I^{n}$. Suppose $w(A)=1$, say $A=f_{ij}^{a},~a\in N$, and $\rho =(x_1,x_2,\ldots ,x_n)$. Then $(0,\ldots ,ax_j,\ldots , 0)=f_{ij}^{a}(x_1,x_2,\ldots ,x_n)\notin I^{n}$ and $(b_1,b_2,\ldots ,b_n)\notin I^{n}$, yield $ax_j\notin I$ and $b_{k}\notin I$, for some k. Since I is weakly prime, by Proposition 2.2, we have $\langle 0\rangle \ne a\langle b_{k}\rangle \nsubseteq I.$ Then there exists $b\in \langle b_{k}\rangle$ such that $0\ne ab\notin I$. Now,

$$\begin{aligned} (0,\ldots ,0)&\ne (0,\ldots , ab,\ldots ,0)\\&=f_{ij}^{ab}(1,\ldots ,1)\\&=f_{ij}^{a}f_{jj}^{b}(1,\ldots ,1)~\text {(}([14,\,Lemma\,3.1(3))]\\&=f_{ij}^{a}(0,\ldots ,b,\ldots ,0). \end{aligned}$$

Since $f_{ij}^{a}\in \mathcal {A}$, and $(0,\ldots ,b,\ldots ,0)\in \langle b_{k}\rangle ^{n}\subseteq \langle (0,\ldots ,b_k,\ldots ,0)\rangle$ by Lemma 3.3$\subseteq \langle (b_1,\ldots ,b_n)\rangle$, it follows that $(\bar{0})\ne \mathcal {A}B^n\nsubseteq I^{n}$. We use the induction on the complexity of A. Assume the induction hypothesis: for all $A\in \mathcal {A}$ and $w(A)<n$, and now consider $w(A)=n$. Then $A=C+D$ or $A=CD$, where $w(C),w(D)<n$. Let $A=C+D$. Now, $(\bar{0})\ne AB^n=(C+D)B^n =CB^n+DB^n\notin I^n+I^n=I^n,~~\text {by induction hypothesis.}$ Therefore, $(\bar{0})\ne \mathcal {A}B^{n}\nsubseteq I^{n}$. Let $A=CD$. Then $(\bar{0})\ne AB^n=(CD)B^n=C(DB^n)$ $\notin I^n$, by induction hypothesis. Therefore, $(\bar{0})\ne \mathcal {A}B^{n}\nsubseteq I^{n}$. $\square$

Proposition 3.6

If $\mathcal {I}$ is a weakly prime ideal in $M_{n}(N)$-group $N^{n} ,$ then $\mathcal {I}_{**}$ is a weakly prime ideal in $_{N}^{}{N}$.

Proof

Suppose that $\mathcal {I}$ is weakly prime $N^{n}.$ By ([9], Lemma 1.3), $\mathcal {I}_{**}$ is an ideal of $_{N}^{}{N}.$ To prove $\mathcal {I}_{**}$ is weakly prime, let A and B be ideals of $_{N}^{}{N}$ with $A \nsubseteq \mathcal {I}_{**}$ and $B \nsubseteq \mathcal {I}_{**}.$ Then there exist $a \in A \backslash \mathcal {I}_{**}$ and $b \in B \backslash \mathcal {I}_{**}$. By ([9], Lemma 1.2), $\left( a,0,\ldots,0\right) \notin \mathcal {I}$ and $\left( b,0,\ldots,0\right) \notin \mathcal {I}$. That is, $f_{11}^{a}\left( 1,\ldots,1\right) \notin \mathcal {I}$ and $\left( b,0,\ldots,0\right) \notin \mathcal {I}.$ Then, $f_{11}^{a}N^{n} \nsubseteq \mathcal {I}$ and $\left( b,0,\ldots,0\right) \notin \mathcal {I}.$ Since $\mathcal {I}$ is weakly prime, we have $(0)\ne f_{11}^{a}\langle \left( b,0,\ldots,0\right) \rangle \nsubseteq \mathcal {I}.$ Then there exists $\left( b',0,\ldots,0\right) \in \langle \left( b,0,\ldots,0\right) \rangle$ such that

$$\begin{aligned} \bar{0}&\ne f_{11}^{a}\left( b',0,\ldots,0\right) \\&=\left( ab',0,\ldots,0\right) \\&\notin \mathcal {I}. \end{aligned}$$

Hence, by ([9], Lemma 1.2), $0\ne ab'\notin \mathcal {I}_{**}$, where $a\in A$ and $\left( b',0,\ldots,0\right) \in \langle \left( b,0,\ldots,0\right) \rangle =\langle b\rangle ^{n}\subseteq B^{n}$, yields $b'\in B$. Therefore, $(0)\ne AB\nsubseteq \mathcal {I}_{**}$, and proves $\mathcal {I}_{**}$ is weakly prime in $_{N}^{}{N}$. $\square$

Definition 3.7

An ideal $\mathcal {I}$ of $N^{n}$ with $M_n(N)N^{n}\nsubseteq \mathcal {I}$, is said to be weakly c-prime if for any $A\in M_n(N)$ and $\rho \in N^{n}$, $\bar{0}\ne A\rho \notin \mathcal {I}$, then $AN^{n}\subseteq \mathcal {I}$ or $\rho \in \mathcal {I}$.

Proposition 3.8

If I is a weakly c-prime ideal of $_{N}^{}{N},$ then $I^{n}$ is weakly c-prime in $M_{n}(N)$-group $N^{n}.$

Proof

Let I be a weakly c-prime ideal of $_{N}^{}{N}$. By ([1], Proposition 4.1), $I^{n}$ is an ideal of $N^{n}$. To prove, $I^{n}$ is weakly c-prime in $N^{n}$, let $A \in M_n(N)$ and $\left( b_1,\ldots ,b_n\right) \in N^{n}$ such that $AN^{n}\nsubseteq I^{n}$ and $\left( b_1,\ldots ,b_n\right) \notin I^n$. Let $w(A)=1$, say $A=f_{ij}^{a},~a\in N$ such that $f_{ij}^{a}(x_1,x_2,\ldots ,x_n)\notin I^{n}$, for some $(x_1,x_2,\ldots ,x_n)\in N^{n}$, and $(b_1,b_2,\ldots ,b_n)\notin I^{n}$. This implies $(0,\ldots ,ax_j,\ldots , 0)\notin I^{n}$ and $(b_1,b_2,\ldots ,b_n)\notin I^{n}$. Then $ax_j\notin I$ and $b_{k}\notin I$ for some k. Since I is weakly c-prime, $0\ne a b_{k} \notin I.$ Hence,

$$\begin{aligned} (0,\ldots , 0,\ldots ,0)&\ne (0,\ldots ,ab_{k},\ldots ,0)\\&=f_{ij}^{ab_{k}}(1,\ldots ,1)\\&=f_{ij}^{a}f_{jk}^{b_{k}}(1,\ldots ,1)\text {}([14, Lemma\,3.1(3))]\\&=f_{ij}^{a}(b_1,\ldots ,b_{k},\ldots ,b_n)\notin I^{n}. \end{aligned}$$

Therefore, $\bar{0}\ne A(b_1,\ldots ,b_n)\notin I^{n}$. The rest of the proof follows by induction on the complexity of A as in the Proposition 3.5. $\square$

Proposition 3.9

If $\mathcal {I}$ is a weakly c-prime ideal of $M_{n}(N)$-group $N^{n},$ then $\mathcal {I}_{**}$ is a weakly c-prime ideal of $_{N}^{}N$.

Proof

Suppose that $\mathcal {I}$ is a weakly c-prime ideal of $N^{n}.$ In view of ([9], Lemma 1.3), it is enough to show the weakly primeness of $\mathcal {I}_{**}$. Let $a,b\in N$ such that $a\notin \mathcal {I}_{**}$ and $b \notin \mathcal {I}_{**}.$ Then, by ([9], Lemma 1.2), $\left( a,0,\ldots,0\right) \notin \mathcal {I}$ and $\left( b,0,\ldots,0\right) \notin \mathcal {I}$. This implies $f_{11}^{a}\left( 1,\ldots,1\right) \notin \mathcal {I}$ and $\left( b,0,\ldots,0\right) \notin \mathcal {I}.$ By taking $A=f_{11}^{a}$, we get $AN^{n}\nsubseteq \mathcal {I}$ and $\left( b,0,\ldots,0\right) \notin \mathcal {I}.$ Since $\mathcal {I}$ is a weakly c-prime ideal of $N^{n}$, we have $\bar{0}\ne A\left( b,0,\ldots,0\right) \notin \mathcal {I}.$ So $\bar{0}\ne f_{11}^{a}\left( b,0,\ldots,0\right) =\left( ab,0,\ldots,0\right) \notin \mathcal {I}$. Thus, by ([9], Lemma 1.2), $0\ne ab\notin \mathcal {I}_{**}$. $\square$

Definition 3.10

An ideal $\mathcal {I}$ of $N^{n}$ with $M_n(N)N^{n}\nsubseteq \mathcal {I}$, is said to be weakly 3-prime if for any $A\in M_n(N)$ and $\rho \in N^{n}$ with $(\bar{0})\ne AM_n(N)\rho \subseteq \mathcal {I}$, then $AN^{n}\subseteq \mathcal {I}$ or $\rho \in \mathcal {I}$.

Proposition 3.11

If I is a weakly 3-prime ideal of $_{N}^{}{N},$ then $I^{n}$ is a weakly 3-prime ideal in $M_{n}(N)$-group $N^{n}.$

Proof

Let I be a weakly 3-prime ideal of $_{N}^{}{N}$. By ([1], Proposition 4.1), $I^{n}$ is an ideal of $N^{n}$ . To show $I^{n}$ is weakly 3-prime in $N^{n}$, let $A \in M_n(N)$ and $\left( b_1,\ldots ,b_n\right) \in N^{n}$ such that $AN^{n}\nsubseteq I^{n}$ and $\left( b_1,\ldots ,b_n\right) \notin I^n$. The proof is based on complexity of A. For this, suppose that $w(A)=1$, say $A=f_{ij}^{a},~a\in N$ such that $f_{ij}^{a}(x_1,x_2,\ldots ,x_n)\notin I^{n}$, for some $(x_1,x_2,\ldots ,x_n)\in N^{n}$, implies $(0,\ldots ,ax_j,\ldots , 0)\notin I^{n}$, and since $(b_1,b_2,\ldots ,b_n)\notin I^{n}$, we get $ax_j\notin I$ and $b_{k}\notin I$, for some k. Since I is weakly 3-prime, we have $(0)\ne aN b_{k} \nsubseteq I.$ Then there exists $n_1\in N$ such that $0\ne an_1b_{k}\notin I$. Therefore,

$$\begin{aligned} (0,\ldots ,0)&\ne (0,\ldots , \underbrace{an_1b_{k}}_{ i^{th} },\ldots ,0)\\&=f_{ik}^{an_1}(b_{1},\ldots ,b_{k},\ldots , b_{n})\\&=f_{ij}^{a}f_{jk}^{n_1}(b_1,\ldots ,b_{k},\ldots ,b_n)\notin I^{n},~ \text {as }f_{jk}^{n_1}\in M_n(N). \end{aligned}$$

Therefore, $(\bar{0})\ne AM_n(N)(b_1,\ldots ,b_n)\nsubseteq I^{n}$. The rest of the proof follows by induction on the complexity of A as in the Proposition 3.5. $\square$

Proposition 3.12

If $\mathcal {I}$ is a weakly 3-prime ideal of $M_{n}(N)$-group $N^{n},$ then $\mathcal {I}_{**}$ is a weakly 3-prime ideal of $_{N}^{}{N}$.

Proof

Suppose $\mathcal {I}$ is a weakly 3-prime ideal of $N^{n}.$ By ([9], Lemma 1.3), $\mathcal {I}_{**}$ is an ideal of $_{N}^{}{N}.$ To show $\mathcal {I}_{**}$ is weakly 3-prime, let $a,b\in N$ such that $aN\nsubseteq \mathcal {I}_{**}$ and $b \notin \mathcal {I}_{**}.$ Then, $ax_1\notin \mathcal {I}_{**}$, for some $x_1\in N$ and $b \notin \mathcal {I}_{**}$, and by ([9], Lemma 1.2), $\left( ax_1,0,\ldots,0\right) \notin \mathcal {I}$ and $\left( b,0,\ldots,0\right) \notin \mathcal {I}$, which implies that $f_{11}^{a}\left( x_1,\ldots,x_n\right) \notin \mathcal {I}$ and $\left( b,0,\ldots,0\right) \notin \mathcal {I}.$ Take $A=f_{11}^{a}$. Then $AN^{n}\nsubseteq \mathcal {I}$ and $\left( b,0,\ldots,0\right) \notin \mathcal {I}.$ Since $\mathcal {I}$ is weakly 3-prime, $\bar{0}\ne AM_n(N)\left( b,0,\ldots,0\right) \nsubseteq \mathcal {I}.$ Then there exists $f_{ij}^{n_1}\in M_n(N),~n_1\in N$ such that $\left( 0,0,\ldots,0\right) \ne f_{11}^{a}f_{ij}^{n_1}\left( b,0,\ldots,0\right) \notin \mathcal {I}$. By ([1], Lemma 3.1 (3)), and N is zero-symmetric, we have the following.

If $1\ne i=j$, then

$$\begin{aligned} f_{11}^{a}f_{ij}^{n_1}\left( b,0,\ldots,0\right)&=f_{1j}^{a0}\left( b,0,\ldots,0\right) \\&=f_{1j}^{0}\left( b,0,\ldots,0\right) \\&=\left( 0,0,\ldots,0\right) ,~\text { a contradiction. } \end{aligned}$$

If $1= i\ne j$, then

$$\begin{aligned} f_{11}^{a}f_{ij}^{n_1}\left( b,0,\ldots,0\right)&=f_{1j}^{an_1}\left( b,0,\ldots,0\right) \\&=\left( an_10,0,\ldots,0\right) \\&=\left( 0,0,\ldots,0\right) ,~\text { a contradiction. } \end{aligned}$$

If $1= i=j$, then

$$\begin{aligned} \left( an_1b,0,\ldots,0\right)&=f_{11}^{an_1}\left( b,0,\ldots,0\right) \\&=f_{11}^{a}f_{ij}^{n_1}\left( b,0,\ldots,0\right) \\&\notin \mathcal {I}. \end{aligned}$$

Therefore, by ([9], Lemma 1.2), $0\ne an_1b\notin \mathcal {I}_{**}$, and thus $(0)\ne aNb\nsubseteq \mathcal {I}_{**}$. $\square$

Definition 3.13

An ideal $\mathcal {I}$ of $N^{n}$ with $M_n(N)N^{n}\nsubseteq \mathcal {I}$, is said to be weakly equiprime (weakly e-prime) if $A\in M_n(N)$ and $\rho _1,\rho _2\in N^{n}$ with $\bar{0}\ne AB\rho _1-AB\rho _2\in \mathcal {I}$, for all $B\in M_n(N)$, then $AN^{n}\subseteq \mathcal {I}$ or $\rho _1-\rho _2\in \mathcal {I}$.

Proposition 3.14

If I is a weakly e-prime ideal of $_{N}^{}{N}$, then $I^{n}$ is a weakly e-prime ideal in $M_{n}(N)$-group $N^{n}.$

Proof

Let I be a weakly e-prime ideal of $_{N}^{}{N}$. By ([1], Proposition 4.1), $I^{n}$ is an ideal of $N^{n}$. To prove $I^{n}$ is weakly e-prime, let $M_n(N) N^n\nsubseteq I^n$ and $A\in M_n(N)$, $\rho _1=(x_1,\ldots ,x_n)$, $\rho _2=(y_1,\ldots ,y_n)\in N^n$ such that $AN^n\nsubseteq I^n$ and $\rho _1-\rho _2\notin I^n$. Suppose that $w(A)=1$, say $A=f_{ij}^{a},a\in N$ such that $f_{ij}^{a}(z_1,\ldots ,z_n)\notin I^n$ for some $(z_1,\ldots ,z_n)\in N^n$ and $\rho _1-\rho _2=(x_1-y_1,\ldots ,x_n-y_n)\notin I^n$. This means that,

$$\begin{aligned} (0,\ldots ,\underbrace{az_j}_{i^{th}},\ldots , 0)\notin I^{n}~ \text {and} ~(x_1-y_1,\ldots ,x_n-y_n)\notin I^n, \end{aligned}$$

and yield $az_j\notin I$ and $x_k-y_k\notin I$, for some k. That is, $aN\nsubseteq I$ and $x_k-y_k\notin I$. Since I is weakly e-prime, we have $0\ne an_1x_k-an_1y_k\notin I$, where $n_1\in N$. Then,

$$\begin{array}{ccccc} (0, \ldots ,0) & \ne (0, \ldots ,\underbrace {a{n_1}{x_k} - a{n_1}{y_k}}_{{i^{th}}}, \ldots ,0)\\ \ne (0, \ldots ,\underbrace {a{n_1}{x_k}}_{{i^{th}}}, \ldots ,0) - (0, \ldots ,\underbrace {a{n_1}{y_k}}_{{i^{th}}}, \ldots ,0)\\ = f_{ik}^{a{n_1}}{\rho _1} - f_{ik}^{a{n_1}}{\rho _2}\\ = f_{ij}^af_{jk}^{{n_1}}{\rho _1} - f_{ij}^af_{jk}^{{n_1}}{\rho _2}([14]),\;{\text{Lemma 3.1(3))}}\\ \notin {I^n}. \end{array}$$

Hence, $\bar{0}\ne AB\rho _1-AB\rho _2\notin I^n$, with $A=f_{ij}^{a}$, $B=f_{jk}^{n_1}$. The rest of the proof follows by induction on the complexity of A as in the Proposition 3.5. $\square$

Proposition 3.15

If $\mathcal {I}$ is a weakly e-prime ideal of an $M_{n}(N)$-group $N^{n},$ then $\mathcal {I}_{**}$ is a weakly e-prime ideal of $_{N}^{}{N}$.

Proof

Suppose that $\mathcal {I}$ is a weakly e-prime ideal of $N^{n}.$ By ([9], Lemma 1.3), we have $\mathcal {I}_{**}$ is an ideal of $_{N}^{}{N}.$ To prove $\mathcal {I}_{**}$ is weakly e-prime , let $a, x,y\in N$ such that $aN\nsubseteq \mathcal {I}_{**}$ and $x-y\notin \mathcal {I}_{**}.$ Then there exists $m_1\in N$ such that $am_1\notin \mathcal {I}_{**}$ and $x-y\notin \mathcal {I}_{**}$. By ([9], Lemma 1.2), we have $\left( am_1,0,\ldots,0\right) \notin \mathcal {I}$ and $\left( x-y,0,\ldots,0\right) \notin \mathcal {I}$. That is,

$$\begin{aligned} f_{11}^{a}\left( m_1,\ldots,m_n\right) =\left( am_1,0,\ldots,0\right) \notin \mathcal {I}~ \text {and} ~ \left( x,0,\ldots,0\right) -\left( y,0,\ldots,0\right) \notin \mathcal {I}. \end{aligned}$$

Take $A=f_{11}^{a}\in M_n(N)$, and $\rho _1=\left( x,0,\ldots,0\right)$, $\rho _2=\left( y,0,\ldots,0\right)$. Then $AN^{n}\nsubseteq \mathcal {I}$ and $\rho _1-\rho _2\notin \mathcal {I}.$ Since $\mathcal {I}$ is weakly e-prime, there exist $B\in M_n(N)$ such that $\bar{0}\ne AB\rho _1-AB\rho _2\notin \mathcal {I}$. Let $B=f_{ij}^{n'},~n'\in N$. Then,

$$\begin{aligned} \bar{0}\ne f_{11}^{a}f_{ij}^{n'}\rho _1-f_{11}^{a}f_{ij}^{n'}\rho _2\notin \mathcal {I}. \end{aligned}$$

By ([1], Lemma 3.1(3)), and N is zero-symmetric, we have the following.

If $1\ne i=j$, then

$$\begin{aligned} f_{11}^{a}f_{ij}^{n'}\rho _1-f_{11}^{a}f_{ij}^{n'}\rho _2&=f_{1j}^{a0}\rho _1-f_{1j}^{a0}\rho _2\\&=f_{1j}^{0}\rho _1-f_{1j}^{0}\rho _2\\&=\left( 0,0,\ldots,0\right) ,~\text { a contradiction.} \end{aligned}$$

If $1= i\ne j$, then

$$\begin{aligned} f_{11}^{a}f_{ij}^{n'}\rho _1-f_{11}^{a}f_{ij}^{n'}\rho _2&=f_{1j}^{an'}\left( x,0,\ldots,0\right) _1-f_{1j}^{an'}\left( y,0,\ldots,0\right) \\&=\left( an'0,0,\ldots,0\right) -\left( an'0,0,\ldots,0\right) \\&=\left( 0,0,\ldots,0\right) ,~\text { a contradiction.} \end{aligned}$$

Now, in case $1= i=j$, by ([1], Lemma 3.1(3)), it follows that

$$\begin{aligned} \bar{0}&\ne f_{11}^{a}f_{ij}^{n'}\rho _1-f_{11}^{a}f_{ij}^{n'}\rho _2\\&=f_{11}^{an'}\rho _2-f_{11}^{an'}\rho _2\\&=\left( an'x,0,\ldots,0\right) -\left( an'y,0,\ldots,0\right) \\&\notin \mathcal {I}. \end{aligned}$$

Therefore, $0\ne an'x-an'y\notin \mathcal {I}_{**}$, proves $\mathcal {I}_{**}$ is weakly e-prime. $\square$

We establish a one-one correspondence between the set of all weakly $\tau$-prime ($\tau =0,c,3,e$), (resp. IFP, (i, 2)-absorbing ($i\in \{c,3\}$)) ideals of $_{N}^{}{N}$ and those of $M_{n}(N)$-group $N^{n}.$

Theorem 3.16

There is a one-one correspondence between the set of all weakly $\tau$-prime ($\tau =0,c,3,e$) ideals of $_{N}^{}{N}$ and those of $M_{n}(N)$-group $N^{n}.$

Proof

Let $\tau \in \{0,c,3,e\}$. Write ${\mathcal {P}} =\{I\trianglelefteq _{N}N:\text {I is weakly} \,\tau \text {-prime}\}$ and $\mathcal {Q}=\{{\mathcal {I}}\trianglelefteq _{M_n(N)}N^{n}:{\mathcal {I}} \hbox { is weakly } \tau \hbox {-prime}\}$. Define

$$\begin{aligned} \varPhi :\mathcal {P}\rightarrow \mathcal {Q} ~ \text {by} ~\varPhi (\textit{I})=\textit{I}^{n}, ~\text {and} ~ \psi :\mathcal {Q} \rightarrow \mathcal {P}~ \text {by}~ \psi (\mathcal {I})=\mathcal {I}_{**}. \end{aligned}$$

Then $\varPhi (I)$ and $\psi (\mathcal {I})$ are weakly $\tau$-prime ideals of $N^n$ and $_{N}^{}{N}$ respectively, as proved above. Now

$$\begin{aligned} \left( \varPhi \circ \psi \right) (\mathcal {I})&=\varPhi (\mathcal {I}_{**})\\&=\left( \mathcal {I}_{**}\right) ^{n}\\&=\mathcal {I}~ \text {(}([10,\,Lemma\,1.5(i)) ], \end{aligned}$$

and

$$\begin{aligned} \left( \psi \circ \varPhi \right) (\textit{I})&=\psi \left( I^{n}\right) \\&=\left( I^{n}\right) _{**}\\&=I~\text {(}([10,\,Lemma\,1.4(iii))]. \end{aligned}$$

Therefore, $\left( \varPhi \circ \psi \right) =id_{\mathcal {Q}}$, and $\left( \psi \circ \varPhi \right) =id_{\mathcal {P}}$, proves that $\mathcal {P}$ and $\mathcal {Q}$ are isomorphic. $\square$

Definition 3.17

An ideal I of G is said to have weakly insertion of factors property (denoted as, weakly IFP), if for any $a\in N$, $g\in G$ with $0\ne ag\in I$, then $ang\in I$, for all $n\in N$.

Example 3.18

([13], Sonata Table no. 134) Consider $N=((\mathbb {Z}_4\times \mathbb {Z}_2),+,\cdot )$ and $G=(N,+)$. Then G is an N-group. Then $\{0\}$ has weakly IFP, but not IFP, since, $7\cdot 4=0$, but $7N4=\{0,2\}\nsubseteq \{0\}$.

Definition 3.19

An ideal $\mathcal {I}$ of $N^n$ is said to have weakly IFP, if for any $A\in M_n(N)$ and $\rho \in N^{n}$ with $(0)\ne A\rho \in \mathcal {I}$, then $AB\rho \in \mathcal {I}$, for all $B\in M_n(N)$.

Lemma 3.20

If I has weakly IFP in $_{N}^{}{N}$, then $I^{n}$ has weakly IFP in $M_{n}(N)$-group $N^{n}$.

Proof

Assume that I has weakly IFP, and let $AB\rho \notin I^{n}$, for some $A,B\in M_n(N)$, $\rho \in N^n$. Let $A=f_{ij}^{a}$, $B=f_{kj}^{b}$, $\rho =(x_1,\ldots ,x_n)$. Then

$f_{ij}^{a}f_{kj}^{b}(x_1,\ldots ,x_n)\notin I^{n}$. By ([1], Lemma 3.1(3)), and N is zero-symmetric, we have the following.

If $j\ne k$, then

$$\begin{aligned} f_{ij}^{a}f_{kj}^{b}(x_1,\ldots ,x_n)&=f_{ij}^{a0}(x_1,\ldots ,x_n)\\&=f_{ij}^{0}(x_1,\ldots ,x_n)\\&=(0,\ldots ,0), \end{aligned}$$

a contradiction.

If $j=k$, then

$$\begin{aligned} (0,\ldots ,\underbrace{abx_j}_{i^{th}},\ldots ,0)&=f_{ij}^{ab}(x_1,\ldots ,x_n)\\&=f_{ij}^{ab}f_{kj}^{b}(x_1,\ldots ,x_n)\\&\notin I^{n}. \end{aligned}$$

This implies $abx_j\notin I$. Since I has weakly IFP, we have $0\ne ax_{j}\notin I$. Therefore, $\bar{0}\ne (0,\ldots ,ax_j,\ldots ,0)=f_{ij}^{a}(x_1,\ldots ,x_n)\notin I^{n}$, yields $\bar{0}\ne A\rho \notin I^{n}$. $\square$

Lemma 3.21

If $\mathcal {I}$ has weakly IFP in $M_n(N)$-group $N^n$, then $\mathcal {I}_{**}$ has weakly IFP in $_{N}^{}{N}$.

Proof

Assume that $\mathcal {I}$ has weakly IFP, and suppose $a,g,n'\in N$ such that $an'g\notin \mathcal {I}_{**}$. Then by ([9], Lemma 1.2), we have $(an'g,\ldots ,0)\notin \mathcal {I}$. Now $f_{11}^{a}f_{11}^{n'}(g,\ldots ,0)=f_{11}^{an'}(g,\ldots ,0)\notin \mathcal {I}$. Taking $A=f_{11}^{a}$, $B=f_{11}^{n'}$ and $\rho =(g,0,\ldots ,0)$ we get $AB\rho \notin \mathcal {I}$. Since $\mathcal {I}$ has weakly IFP, we have $\bar{0}\ne A\rho \notin \mathcal {I}$. This implies, $\bar{0}\ne f_{11}^{a}(g,\ldots ,0)=(ag,\ldots ,0)\notin \mathcal {I}$, and get $0\ne ag\notin \mathcal {I}_{**}$. $\square$

From Lemma 3.20, Lemma 3.21, ([9], Lemma 1.5(i), 1.4(iii)), we get

Theorem 3.22

There is a one-one correspondence between the set of all weakly IFP ideals of $_{N}^{}{N}$ and those of $M_{n}(N)$-group $N^{n}.$

Proof

Let $\mathcal {P}=\{I\trianglelefteq _{N}N:I\hbox { has weakly IFP}\}$ and

$\mathcal {Q}=\{\mathcal {I}\trianglelefteq _{M_n(N)}N^{n}:{\mathcal {I}}\hbox { has weakly IFP}\}$. Define

$$\begin{aligned} \varPhi :\mathcal {P}\rightarrow \mathcal {Q} ~\text { by }~ \varPhi (I)=I^{n} . \end{aligned}$$

By Lemma 3.20, $\varPhi (I)$ has IFP. Define

$$\begin{aligned} \psi :\mathcal {Q} \rightarrow \mathcal {P}~ \text { by }~ \psi (\mathcal {I})=\mathcal {I}_{**}. \end{aligned}$$

By Lemma 3.21, $\psi (\mathcal {I})$ has IFP. Now $\left( \varPhi \circ \psi \right) (\mathcal {I}) =\varPhi (\mathcal {I}_{**})=\left( \mathcal {I}_{**}\right) ^{n}=\mathcal {I}$ ([9], Lemma 1.5(i)) and $\left( \psi \circ \varPhi \right) (I) =\psi \left( I^{n}\right) =\left( I^{n}\right) _{**}=I$ ([9], Lemma 1.4(iii)). Therefore, $\left( \varPhi \circ \psi \right) =id_{\mathcal {Q}}$, and $\left( \psi \circ \varPhi \right) =id_{\mathcal {P}}$. Hence, $\mathcal {P}$ and $\mathcal {Q}$ are isomorphic. $\square$

We introduce the notions weakly (c, 2)-absorbing ideal and weakly (3, 2)-absorbing ideal as generalizations of c-prime and 3-prime ideal of an N-group G, respectively. We establish a one-one correspondence between (i, 2), $i\in \{c,3\}$-absorbing ideals of N (over N) and those of $N^{n}$. Tapatee et. al [14] introduced completely 2-absorbing (abbr. (c, 2)-absorbing) ideals in G and derived the properties such as homomorphic images, inverse images of (c, 2)-absorbing ideals.

Definition 3.23

Let $I\lhd _{N}G$ such that $NG\nsubseteq I$. Then

• 1. I is weakly (c, 2)-absorbing if for $a,b\in N$, $g\in G$, $0\ne abg\in I$, then $abG\subseteq I$ or $ag\in I$ or $bg\in I$.

• 2. I is weakly (3, 2)-absorbing if for $a,b\in N$, $g\in G$, $(0)\ne abNg\subseteq I$, then $abG\subseteq I$ or $ag\in I$ or $bg\in I$.

Example 3.24

Consider $N=(A_4,+,\cdot )$ and $G=N$, given in ([11], Table O(1)). Then G is an N-group, where $+$ and $\cdot$ is defined as follows:

+	0	1	2	3	4	5	6	7	8	9	10	11
0	0	1	2	3	4	5	6	7	8	9	10	11
1	1	0	3	2	7	6	5	4	10	11	8	9
2	2	3	0	1	5	4	7	6	11	10	9	8
3	3	2	1	0	6	7	4	5	9	8	11	10
4	4	5	6	7	8	9	10	11	0	1	2	3
5	5	4	7	6	11	10	9	8	2	3	0	1
6	6	7	4	5	9	8	11	10	3	2	1	0
7	7	6	5	4	10	11	8	9	1	0	3	2
8	8	9	10	11	0	1	2	3	4	5	6	7
9	9	8	11	10	3	2	1	0	6	7	4	5
10	10	11	8	9	1	0	3	2	7	6	5	4
11	11	10	9	8	2	3	0	1	5	4	7	6

$\cdot$	0	1	2	3	4	5	6	7	8	9	10	11
0	0	0	0	0	0	0	0	0	0	0	0	0
1	0	0	0	0	0	0	0	0	0	0	0	0
2	0	0	0	0	0	0	0	0	0	0	0	0
3	0	0	0	0	0	0	0	0	0	0	0	0
4	0	0	0	0	0	0	0	0	4	4	4	6
5	0	0	0	0	0	0	0	0	4	4	4	6
6	0	0	0	0	0	0	0	0	4	4	4	6
7	0	0	0	0	0	0	0	0	4	4	4	6
8	0	0	0	0	0	0	0	0	8	8	8	11
9	0	0	0	0	0	0	0	0	8	8	8	11
10	0	0	0	0	0	0	0	0	8	8	8	11
11	0	0	0	0	0	0	0	0	8	8	8	11

The ideals are $I_1=\{0\}$ and $I_2=\{0,1,2,3\}$. Then $I_2$ is not a weakly 3-prime ideal of G, since $4N5=\{0\}\in I_2$, but $4G\nsubseteq I_2$ and $5\notin I_2$ whereas $I_2$ is weakly (3, 2)-absorbing ideal of G.

Definition 3.25

Let $\mathcal {I}\lhd _{M_n(N)}N^{n}$ such that $M_n(N)N^{n}\nsubseteq \mathcal {I}$. Then

• 1. $\mathcal {I}$ is weakly (c, 2)-absorbing if for $A,B\in M_n(N)$ and $\rho \in N^{n}$, $\bar{0}\ne AB\rho \in \mathcal {I}$, then $ABN^{n}\subseteq \mathcal {I}$ or $A\rho \in \mathcal {I}$ or $B\rho \in \mathcal {I}$.

• 2. $\mathcal {I}$ is weakly (3, 2)-absorbing if for $A,B\in M_n(N)$ and $\rho \in N^{n}$, $(\bar{0})\ne ABM_n(N)\rho \subseteq \mathcal {I}$, then $ABN^{n}\subseteq \mathcal {I}$ or $A\rho \in \mathcal {I}$ or $B\rho \in \mathcal {I}$.

Proposition 3.26

If I is a weakly (c, 2)-absorbing ideal of $_{N}^{}{N},$ then $I^{n}$ is a weakly (c, 2)-absorbing ideal in $M_{n}(N)$-group $N^{n}.$

Proof

Let I be a weakly (c, 2)-absorbing ideal of $_{N}^{}{N}$. By ([1], Proposition 4.1), $I^{n}$ is an ideal of $N^{n}$. To show $I^{n}$ is weakly (c, 2)-absorbing in $N^{n}$, let $A, B\in M_n(N)$ and $\rho \in N^n$ such that $\bar{0}\ne AB\rho \in I^{n}$. Suppose that $w(A)=w(B)=1$, say $A=f_{11}^{a},$, $B=f_{11}^{b},~a,b\in N$ and $\rho =(x_1,\ldots ,x_n)$. Then by ([1], Lemma 3.1 (3)), we have

$$\begin{aligned} \bar{0}\ne&f_{11}^{a}f_{11}^{b}(x_1,\ldots ,x_n)\\&=f_{11}^{ab}(x_1,\ldots ,x_n)\\&=(abx_1,\ldots ,0)\\&\in I^n. \end{aligned}$$

This implies, $0\ne abx_1\in I$. Since I is weakly (c, 2)-absorbing in $_{N}^{}{N}$, we have $abN\subseteq I$ or $ax_1\in I$ or $bx_1\in I$, and so $abx_i\in I$, for all i. Thus,

$$\begin{aligned} (abx_1,0,\ldots ,0)\in I^{n}~ \text {or}~(ax_1,\ldots , 0)\in I^{n}~ \text {or}~(bx_1,\ldots , 0)\in I^n. \end{aligned}$$

Now,

$$\begin{aligned} f_{11}^{a}f_{11}^{b}(x_1,x_2,\ldots ,x_n)&=f_{11}^{ab}(x_1,x_2,\ldots ,x_n)\\&=(abx_1,0,\ldots ,0)\\&\in I^{n}, ~\text {for all }\,x_i\in N \end{aligned}$$

or

$$\begin{aligned} f_{11}^{a}(x_1,x_2,\ldots ,x_n)\in I^{n}~ \text {or}~ f_{11}^{b}(x_1,x_2,\ldots ,x_n)\in I^{n}, \end{aligned}$$

implies that $ABN^{n}\subseteq I^{n}$ or $A\rho \in I^{n}$ or $B\rho \in I^{n}$. $\square$

Proposition 3.27

If $\mathcal {I}$ is a weakly (c, 2)-absorbing ideal of $M_{n}(N)$-group $N^{n},$ then $\mathcal {I}_{**}$ is a weakly (c, 2)-absorbing ideal of $_{N}^{}{N}$.

Proof

Suppose that $\mathcal {I}$ is a weakly (c, 2)-absorbing ideal of $N^{n}.$ By ([9], Lemma 1.3), $\mathcal {I}_{**}$ is an ideal of $_{N}^{}{N} .$ To prove $\mathcal {I}_{**}$ is a weakly (c, 2)-absorbing ideal of $_{N}^{}{N}$, let $a,b,c\in N$ such that $0\ne abc\in \mathcal {I}_{**}$. Then by ([9], Lemma 1.2), we have $0\ne (abc,0,\ldots ,0)\in \mathcal {I}$. This implies that,

$$\begin{aligned} \bar{0}\ne (abc,0,\ldots ,0)&=f_{11}^{ab}(c,0,\ldots ,0)\\&=f_{11}^{a}f_{11}^{b}(c,0,\ldots ,0)\\&\in \mathcal {I}, \end{aligned}$$

shows that $\bar{0}\ne AB\rho _1\in \mathcal {I}$, where $A=f_{11}^{a},~B=f_{11}^{b},\rho _1=(c,0,\ldots ,0)$. Since $\mathcal {I}$ is a weakly (3, 2)-absorbing ideal of $N^{n}$, we get $ABN^{n}\subseteq \mathcal {I}$ or $A\rho _1\in \mathcal {I}$ or $B\rho _1\in \mathcal {I}$. Hence, $f_{11}^{a}f_{11}^{b}(x_1,\ldots ,x_n)\in \mathcal {I}$, for all $(x_1,\ldots ,x_n)\in N^{n}$ or $f_{11}^{a}(c,\ldots , 0)\in \mathcal {I}$ or $f_{11}^{b}(c,\ldots ,0)\in \mathcal {I}$, implies that $(abx_1,\ldots ,0)\in \mathcal {I}$ or $(ac,\ldots , 0)\in \mathcal {I}$ or $(bc,\ldots , 0)\in \mathcal {I}$. Then, by ([9], Lemma 1.2), we have $abx_1\in \mathcal {I}_{**}$, for all $x_1\in N$ or $ac\in \mathcal {I}_{**}$ or $bc\in \mathcal {I}_{**}$, and so $abN\subseteq \mathcal {I}_{**}$ or $ac\in \mathcal {I}_{**}$ or $bc\in \mathcal {I}_{**}$. $\square$

Proposition 3.28

If I is a weakly (3, 2)-absorbing ideal of $_{N}^{}{N}$, then $I^{n}$ is a weakly (3, 2)-absorbing ideal of $M_{n}(N)$-group $N^{n}$.

Proof

Let I be a weakly (3, 2)-absorbing ideal of $_{N}^{}{N}$. By ([1], Proposition 4.1), $I^{n}$ is an ideal of $N^{n}$. To show $I^{n}$ is weakly (3, 2)-absorbing in $N^{n}$, let $A, B\in M_n(N)$ and $\rho \in N^n$ such that $(\bar{0})\ne ABM_n(N)\rho \subseteq I^{n}$. Then, $\bar{0}\ne ABC\rho \in I^{n}$, for every $C\in M_n(N)$. Suppose that $w(A)=w(B)=w(C)=1$, say $A=f_{11}^{a},$, $B=f_{11}^{b}$, $C=f_{11}^{c},~a,b,c\in N$ and $\rho =(x_1,\ldots ,x_n)$. Then by ([1], Lemma 3.1 (3)), we have

$$\begin{aligned} \bar{0}\ne&f_{11}^{a}f_{11}^{b}f_{11}^{c}(x_1,\ldots ,x_n)\\&=f_{11}^{abc}(x_1,\ldots ,x_n)\\&=(abcx_1,\ldots ,0)\\&\in I^n. \end{aligned}$$

This implies, $0\ne abcx_1\in I$, for all $c\in N$. Hence, $(0)\ne abNc_1\subseteq I$. Since I is weakly (3, 2)-absorbing in $_{N}^{}{N}$, we have $abN\subseteq I$ or $ax_1\in I$ or $bx_1\in I$, and so $abx_i\in I$, for all i. Thus,

$$\begin{aligned} (abx_1,0,\ldots ,0)\in I^{n}~ \text {or}~(ax_1,\ldots , 0)\in I^{n}~ \text {or}~(bx_1,\ldots , 0)\in I^n. \end{aligned}$$

Now,

$$\begin{aligned} f_{11}^{a}f_{11}^{b}(x_1,x_2,\ldots ,x_n)&=f_{11}^{ab}(x_1,x_2,\ldots ,x_n)\\&=(abx_1,0,\ldots ,0)\\&\in I^{n}, ~\text {for all} x_i\in N \end{aligned}$$

or

$$\begin{aligned} f_{11}^{a}(x_1,x_2,\ldots ,x_n)\in I^{n}~ \text {or}~ f_{11}^{b}(x_1,x_2,\ldots ,x_n)\in I^{n}, \end{aligned}$$

implies that $ABN^{n}\subseteq I^{n}$ or $A\rho \in I^{n}$ or $B\rho \in I^{n}$. $\square$

Proposition 3.29

If $\mathcal {I}$ is a weakly (3, 2)-absorbing ideal of $M_{n}(N)$-group $N^{n},$ then $\mathcal {I}_{**}$ is a weakly (3, 2)-absorbing ideal of $_{N}^{}{N}$.

Proof

Suppose that $\mathcal {I}$ is a weakly (3, 2)-absorbing ideal of $N^{n}.$ By ([9], Lemma 1.3), $\mathcal {I}_{**}$ is an ideal of $_{N}^{}{N} .$ To prove $\mathcal {I}_{**}$ is a weakly (3, 2)-absorbing ideal of $_{N}^{}{N}$, let $a,b,x\in N$ such that $abN\nsubseteq \mathcal {I}_{**}$, $ax\notin \mathcal {I}_{**}$ and $bx\notin \mathcal {I}_{**}$. Then there exist $n\in N$ such that $abn\notin \mathcal {I}_{**}$, $ax\notin \mathcal {I}_{**}$ and $bx\notin \mathcal {I}_{**}$. Now by ([9], Lemma 1.2) $\left( abn,\ldots ,0\right) \notin \mathcal {I}$, $\left( ax,\ldots ,0\right) \notin \mathcal {I}$ and $\left( bx,\ldots ,0\right) \notin \mathcal {I}$. Hence, $f_{11}^{ab}(n,\ldots ,0)\notin \mathcal {I}$, $f_{11}^{a}\left( x,0,\ldots ,0\right) \notin \mathcal {I}$ and $f_{11}^{b}\left( x,\ldots ,0\right) \notin \mathcal {I}$. Take $A=f_{11}^{a}$, $B=f_{11}^{b}$ and $\rho =(x,0,\ldots ,0)$. Then $ABN^{n}\nsubseteq \mathcal {I}$, $A\rho \notin \mathcal {I}$ and $B\rho \notin \mathcal {I}$. Since $\mathcal {I}$ is weakly (3, 2)-absorbing in $N^{n}$, we have $(\bar{0})\ne ABM_n(N)\rho \nsubseteq \mathcal {I}$. Then $(\bar{0})\ne ABC\rho \notin \mathcal {I}$, for some $C\in M_n(N)$. Now,

$$\begin{aligned} ABC=f_{11}^{a}\,f_{11}^{b}C&=\,f_{11}^{a}\,f_{11}^{b}Cf_{11}^{e}\\&=\,f_{11}^{a}\,f_{11}^{b}(f_{11}^{r_1}+\ldots +f_{n1}^{r_n}), ~\text {(}([7,\,Lemma\,2.3)]\\&=\,f_{11}^{a}\,f_{11}^{b}f_{11}^{r_1}~\text {(}([14,\,Lemma\,3.1 (5))]\\&=\,f_{11}^{abr_1} ~\text {(}([14,\,Lemma\,3.1 (3))]. \end{aligned}$$

Hence,

$$\begin{aligned} (\bar{0})\ne ABC\rho&=f_{11}^{abr_1}\rho \\&=f_{11}^{abr_1}(x,0,\ldots , 0)\\&=(abr_1x,0,\ldots , 0)\\&\mathcal {I}. \end{aligned}$$

Therefore, by ([9], Lemma 1.2), $0\ne abr_1x\notin \mathcal {I}_{**}$, shows that, $(0)\ne abNx\nsubseteq \mathcal {I}_{**}$. $\square$

The following is a one-one correspondence similar to Theorem 3.16.

Theorem 3.30

There is a one-one correspondence between the set of all weakly (i, 2)-absorbing, $i\in \{c,3\}$ ideals of $_{N}^{}{N}$ and those of $M_{n}(N)$-group $N^{n}.$

Proof

Let $i\in \{c,3\}$ and let $\mathcal {P}=\{I\trianglelefteq _{N}N:I\hbox { is weakly}\, (i,2)\hbox {-absorbing}\}$ and $\mathcal {Q}=\{\mathcal {I}\trianglelefteq _{M_n(N)}N^{n}:\mathcal {I} \text { is weakly }(i,2)\text {-absorbing}\}$. Define $\varPhi :\mathcal {P}\rightarrow \mathcal {Q}$ by $\varPhi (I)=I^{n}$. By Theorem 3.26 and 3.28, $\varPhi (I)$ is weakly (i, 2)-absorbing. Define $\psi :\mathcal {Q} \rightarrow \mathcal {P}$ by $\psi (\mathcal {I})=\mathcal {I}_{**}.$ By Theorem 3.27 and 3.29, $\psi (\mathcal {I})$ is weakly (i, 2)-absorbing. Now $\left( \varPhi \circ \psi \right) (\mathcal {I}) =\varPhi (\mathcal {I}_{**})=\left( \mathcal {I}_{**}\right) ^{n}=\mathcal {I}$ ([9], Lemma 1.5(i)) and $\left( \psi \circ \varPhi \right) (I) =\psi \left( I^{n}\right) =\left( I^{n}\right) _{**}=I$ ([9], Lemma 1.4(iii)). Therefore, $\left( \varPhi \circ \psi \right) =id_{\mathcal {Q}}$, and $\left( \psi \circ \varPhi \right) =id_{\mathcal {P}}$. Hence, $\mathcal {P}$ and $\mathcal {Q}$ are isomorphic. $\square$

4 Conclusion

We have proved the one-one correspondence between the weakly $\tau$-prime ideals $(\tau =0,c,3,e)$ of N-group N and those of $M_n(N)$-group $N^{n}$. Further, we have introduced the notions weakly (c, 2)-absorbing ideals and weakly (3, 2)-absorbing ideals as a generalization of c-prime and 3-prime ideals of N-group G, respectively; and finally obtained the one-one correspondence between (i, 2), $i\in \{c,3\}$-absorbing ideals of N (over N) and those of $M_n(N)$-group $N^{n}$. As a future research work, one can study the radical properties and hyperstructural aspects [15, 16] of weakly prime ideals in $M_n(N)$-group $N^{n}$.