A note on $\phi$-$\lambda$-rings and $\phi$-$\Delta$-rings

Abstract

Let ${\mathcal {H}}$ denotes the set of all commutative rings R in which the set of all nilpotent elements, denoted by $\text {Nil}(R)$, is a prime ideal of R and is comparable to every ideal of R. Let $R\in {\mathcal {H}}$ be a ring and T(R) be its total quotient ring. Then there is a ring homomorphism $\phi : T(R) \rightarrow R_{\text {Nil}(R)}$ defined as $\phi (r/s) = r/s$ for all $r\in R$ and for all non-zero-divisors $s\in R$. A ring $R\in {\mathcal {H}}$ is said to be a $\phi$-$\lambda$-ring if the set of all rings between $\phi (R)$ and $T(\phi (R))$ is linearly ordered by inclusion. If $R_1 + R_2$ is a ring between $\phi (R)$ and $T(\phi (R))$ for each pair of rings $R_1, R_2$ between $\phi (R)$ and $T(\phi (R))$, then R is said to be a $\phi$-$\Delta$-ring. Let $R\in {\mathcal {H}}$ be a $\phi$-$\lambda$-ring and $T\in {\mathcal {H}}$ be a ring properly containing R such that $\text {Nil}(T) = \text {Nil}(R)$. We show that if all but finitely many intermediate rings between R and T are $\phi$-$\lambda$-rings (resp., $\phi$-$\Delta$-rings), then all the intermediate rings are $\phi$-$\lambda$-rings (resp., $\phi$-$\Delta$-rings under some conditions). Moreover, the pair (R, T) is a residually algebraic pair. Two new ring theoretic properties, namely, $\phi$-$\lambda$-property of rings and $\phi$-$\Delta$-property of rings are introduced and studied.