Group inverse for two classes of 2×2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\times 2$$\end{document} block matrices over rings

Let R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R$$\end{document} be an associative ring with unity 1. In this paper, existence of the group inverse of two classes of 2×2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\times 2$$\end{document} block matrices is investigated. We obtain the sufficient and necessary conditions. And further, the representations of the group inverse of two classes are given. (1) M=AX+YBAB0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M=\left( \begin{array}{c@{\quad }l} AX+YB&{}A\\ B&{}0 \end{array}\right) $$\end{document}, where A♯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^{\sharp }$$\end{document} exists, XA=AX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$XA=AX$$\end{document} and X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X$$\end{document} is invertible; (ii) M=ABCD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M=\left( \begin{array}{l@{\quad }l} A&{}B\\ C&{}D\end{array}\right) $$\end{document}, where CA=C,AB=B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CA=C, AB=B$$\end{document} and (D-CB)♯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(D-CB)^{\sharp }$$\end{document} exists. The results extend the early works of Cao and Li (Appl Math Comput 217:10271–10277, 2011) and Liu and Yang (Appl Math Comput 218:8978–8986, 2012). Some examples are given to illustrate our results.

has applications in singular differential and difference equations, Markov chains and iterative methods. For example, Heinig (1997) investigated the group inverse of Sylvester transformation. Wei and Diao (2005) studied the representation of the group inverse of a real singular Toeplitz matrix which arises in scientific computing and engineering. Catral et al. (2008) studied the existence of A . More details and its applications about group (Drazin) inverse can be found in the literature (Soares and Latouche 2002;Bu and Zhang 2011;Golub and Greif 2003;Wei 1998;Ben-Israel and Greville 2003).
In the past few decades, group (Drazin) inverse has been studied extensively. Recently, more and more authors pay close attention to them and a large body of works can be found in literature (see, for example Cao and Zhao 2012;Bu et al. 2009;Bu et al. 2010;Liu et al. 2011;Bu et al. 2012.) However, most of the relative results were obtained over skew fields and complex fields. Recently, Zhang and Bu (2012) and Ge et al. (2012) investigated the group inverse of some 2 × 2 block matrices over right Ore domains. It is well known that a ring is called a right Ore domain if it possesses no zero divisors and every two elements of the ring have a right common multiple, see Zhang and Bu (2012). Skew field and complex field are in the right Ore domain. It is very important to study the group inverse in more general rings. In fact, the concept of the Drazin inverse (group inverse) is based on associative rings, see Drazin (1958). Motivated by the excellent results mentioned above, the aim of this paper is to extend some classical results to even more general setting, i.e., associative rings. The results in this paper improve two following works: Cao and Li (2011) gave the group inverse of the block matrix AX + Y B A B 0 under some conditions over skew fields. We extend the relative results to associative rings. We should pointed here that the method used here is quite different from that of Cao and Li (2011). Liu and Yang (2012) studied the group inverse of the block matrix A B C D under the conditions D = 0, AB = B, C A = C and A = A 2 over complex fields. In this paper, we improve the results to associative rings under more simplify conditions. Next, we give some explanations on notations used in this paper. Let R, , K be an associative ring with unity 1, right Ore domain with unity 1 and skew field, respectively. R m×n , m×n and K m×n denote the set of all m × n matrices over R, and K , respectively. R (m) (R m ) denotes the set of all m-dimensional row (column) vectors over R. Let A ∈ m×n , denote the range and the row range of a matrix A by R(A) and We denote I n − A A by A π . A matrix A is regular if there exists a matrix X such that AX A = A. Define A{1} = {X : AX A = A, and if one needs to pick an element of A{1}, simply we write B ∈ A{1} or A (1) ∈ A{1}. If A ∈ m×n , r (A) denotes the rank of A, see Zhang and Bu (2012).

Some Lemmas
Lemma 1 Let A ∈ R n×n , the followings are equivalent: Proof It follows from Proposition 8.22 of Bhaskara Rao (2002) that (i) ⇔ (ii) holds. Next, we prove the relation of (ii) ⇔ (iii).
Using the fact that the right Ore domain has unity 1, we can easily obtain that R(A) = R(A 2 ) and R r (A) = R r (A 2 ) imply A = A 2 X and A = Y A 2 , respectively. Ge et al. (2012) Let A ∈ n×n , then the followings are equivalent:

Lemma 3 Ge et al. (2012)
Let A ∈ n×m , B ∈ m×n , then the followings are equivalent: Proof By similar arguments of Lemmas 1 and 3, we can easily obtain the results.
Lemma 5 Suppose that A ∈ R m×n , B ∈ R n×m . If (AB) and (B A) exist, then the following equalities hold.
Proof The proof of this Lemma is similar to Lemma 3 in Ge et al. (2012), so we omit it here.

Main results
and By Lemma 1, there exist matrices X and Y over R such that M = M 2 X and Y M 2 = M. Let It follows from Eqs. (1)-(3) and M = M 2 X , we can obtain following equations On the other hand, we have It follows from (1), (8), (9) and Y M 2 = M, we have From (6), we have Substituting it into (4). Obviously, we have Recall that X is invertible and A is group invertible. It follows that and Substitute (14) into (12), we have Substitute (15) into (16), we have In what follows, we give the proof of "if" part. It follows form Lemma 4 that R(BT B) = R(B) and R r (BT B) = R r (B) imply (T B) and (BT ) exist. Let We claim that X 1 , X 2 , X 3 and X 4 satisfy the Eqs. (4)-(7). Next, we verify the claim by computation separatively.
(ii)By Lemmas 1, 4 and 5, the expression of M can be obtained from M = Y M X .
The following corollaries follow from Theorem 1.

T B) A A + (T B) π A + (T B) π A B(T B) A A , M 3 = −B(T B) A + B[(T B) ] 2 A π + B(T B) A B(T B) A π and M 4 = −B(T B) A − B[(T B) ] 2 A A − B(T B) A B(T B) A A .
Proof The proof immediately follows from Theorem 1 if we define X = I, Y = 0. The calculations and conclusions are analogous, and we omit them. Proof The result immediately follows from Lemma 2 and 3. It is easy to verify AX = X A and X is invertible. By a direct computation, we know that A and (T B) exist.

Further, we have
Clearly, R(B) = R(BT B) and R r (B) = R r (BT B). According to Theorem 1, we know that M exists.
The representation of M immediately follows from (ii) of Theorem 1,

Example for Corollary 2: Let
It is easy to verify AX = X A and X is invertible. By a direct computation, we know that A and (T B) exist. We can easily obtain Clearly, R(B) = R(BT B). The representation of M immediately follows from Corollary 2,

if S exists, then (i) M exists if and only if F = A 2 + B S π C is regular and R(F) = R(A), R r (F) = R r (A);
(

ii) If M exists, then M
Proof The "only if" part. Obviously, and and It follows from (17), (18), (19) and M 2 X = M, we can obtain Similarly, it follows from (17), (18), (20) and Y M 2 = M, we can obtain It follows from (21) and (22) that On the other hand, it follows from F = A 2 + B S π C, AB = B and C A = C that we have It is easy to see that F = A(A + B S π C) = A(I + B S π C)A = F X 1 (I + B S π C)Y 1 F, which implies that F is regular.
The "if" part.
Since F is regular, we can take F (1) ∈ F{1}. Let Next we need to prove M = M 2 X and M = Y M 2 hold, i.e., (21) and (22) hold. In fact, from X 3 = 0, X 4 = S , it is easy to verify S 2 X 3 = 0 and S 2 X 4 = S. Note that R(A) = R(F), there exists V ∈ R n×n such that F V = A. Hence, we have and This implies that (21) holds, i.e., It follows (22) This complete the proof of Theorem 2.
In particular, we now give Corollaries and Example of Theorem 2. Remark 1 In fact, Theorem 1 improves the result of Theorem 1.1 in Zhao et al. (2012) to the rings, and Theorem 2 reduces the restrict condition of Corollary 2.2 in Liu and Yang (2012) and improves the relative results to even more general rings.