A Generalization of VNL-Rings and P P -Rings

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1 Journal of Mathematical Research with Applications Mar, 2017, Vol 37, No 2, pp DOI:103770/jissn: A Generalization of VNL-Rings and P P -Rings Yueming XIANG Department of Mathematics and Applied Mathematics, Huaihua University, Hunan , P R China Abstract Let R be a ring An element a of R is called a left P P -element if Ra is projective The ring R is said to be a left almost P P -ring provided that for any element a of R, either a or 1 a is left P P We develop, in this paper, left almost P P -rings as a generalization of von Neumann local VNL rings and left P P -rings Some properties of left almost P P -rings are studied and some examples are also constructed Keywords VNL-rings; left P P -rings; left almost P P -rings MR2010 Subject Classification 16E50; 16L99 1 Introduction As a common generalization of von Neumann regular rings and local rings, Contessa in [1] called a commutative ring R von Neumann local VNL if for each a R, either a or 1 a is von Neumann regular An element a R is von Neumann regular provided that there exists an element x R such that a axa VNL-rings are also exchange rings Some properties of VNL-rings and SVNL-rings were investigated in [2] Later Chen and Tong in [3] defined a noncommutative ring to be a VNL-ring Some results on commutative VNL-rings were extended Moreover, Grover and Khurana in [4] characterized VNL-rings in the sense of relating them to some familiar classes of rings On the other hand, we recall that a ring R is said to be left P P see [5] or left Rickart provided that every principal left ideal is projective, or equivalently the left annihilator of any element of R is a summand of R R A ring is called a P P -ring if it is both left and right P P -ring Examples include von Neumann regular rings and domains The P P -rings and their generalizations have been extensively studied by many authors [5 15] We say that, in this paper, an element a of R is left P P in R if Ra is projective, or equivalently, if l R a Re for some e 2 e R Obviously, R is a left P P -ring if and only if every element of R is left P P A ring R is said to be a left almost P P -ring provided that for any element a of R, either a or 1 a is left P P left almost P P -rings are introduced as the generalization of left P P -rings and VNL-rings Some examples turn out to show that this generalization is non-trivial In Section 2, we investigate the properties of left almost P P -rings Extensions of left P P -rings are considered in Section 3 Received January 5, 2016; Accepted November 23, 2016 Supported by the Natural Science Foundation of Hunan Province Grant No 2016JJ2050 address: xymls999@126com Some results on left P P -rings are

2 200 Yueming XIANG extended onto left almost P P -rings Section 4 focuses on semiperfect, left almost P P -rings We give the structure of this class rings Throughout R is an associative ring with identity and all modules are unitary JR will denote the Jacobson radical of R Z n stands for the ring of integers mod n M n R denotes the ring of all n n matrices over a ring R with an identity I n If X is a subset of R, the left resp, right annihilator of X in R is denoted by l R X resp, r R X If X {a}, we usually abbreviate it to l R a resp, r R a For the usual notations we refer the reader to [1], [7] and [16] 2 Left almost P P -rings We start this section with the definition Definition 21 Let R be a ring and a R a is called a left P P -element in R if Ra is projective, or equivalently, l R a Re for some e 2 e R The ring R is said to be a left almost P P -ring provided that for any element a of R, either a or 1 a is left P P Similarly, right almost P P -rings can be defined A ring R is called almost P P if it is left and right almost P P Remark 22 1 Obviously, left P P -rings are left almost P P -rings 2 Every VNL-ring is a left and right almost P P -ring 3 Clearly, a R is left P P if and only if au is left P P for every unit element u R Example 23 1 The ring Z of integers is an almost P P -ring but not a VNL-ring 2 The ring Z 4 of integers mod 4 is an almost P P -ring but not a P P -ring 3 Let R { a b 0 a a, b Z2 } Then R {, , , 0 1 } If c, let e ; If c or , let e 1 c ; If c 0 1, consider, let e In either case, we have lr c Re or l R 1 c Re So R is a left almost P P -ring choose c 0 1 R, Rc is not projective since lr c JR cannot be generated by an idempotent, then R is not a left P P -ring Example 24 If R is a left P P -ring, S is local and let R M S be bimodule, then R M 0 S is a left almost P P -ring Proof Let T R M 0 S For any α a m 0 b T Since S is local, b or 1S b is invertible Assume that b is invertible Note that a is a left P P -element in R, so there exists e 2 e R such that l R a Re Then e emb 1 a m 0 b Let β e emb 1 Then β 2 β T and T β l T α Now for any a 1 m 1 0 b 1 lt α, a1 m 1 a m 0 b 1 0 b, we have a1 l R a Re, b 1 0 and m 1 a 1 mb 1 So a 1 m 1 0 b 1 e emb 1 T β This implies that α is a left P P -element in T r 0 Assume that 1 S b is invertible As 1 R a is a left P P -element in R, there exists f 2 f R

3 A generalization of VNL-rings and P P -rings 201 such that l R 1 R a Rf Similarly, 1 T α 1 R S a m 0 b 1R a m 0 1 S b is a left P P - element in T Therefore, we complete the proof Now we elaborate some properties of left almost P P -rings Proposition 25 Let R be a left almost P P -ring Then the following results hold Proof 1 The center of R is an almost P P -ring 2 For every e 2 e R, the corner ring ere is a left almost P P -ring 1 Let CR be the center and x CR Since R is left almost P P, x or 1 x is a left P P -element in R If x is a left P P -element, then l R x Re for some e e 2 R Note that l R x Re is an ideal and l R x r R x because x CR It follows that for every r R, er ere re, and hence e CR We now prove that l CR x CRe Clearly, l CR x l R x CR and CRe l CR x Let a l CR x, then a l R x and so a ae CRe Thus, l CR x CRe Consequently l CR x CRe Note that 1 x is also in CR, if 1 x is a left P P -element in R, then 1 x is a left P P -element in CR by using the similar method above Therefore, CR is also an almost P P -ring 2 For any 0 a ere, a or 1 a is left P P in R by hypothesis Assume that a is left P P, then l R a Rf for some f 2 f R Note that l ere a l R a ere and 1 e l R a, so 1 e 1 ef and fe efe Write fe g, then g 2 g ere So ga fea fa 0 On the other hand, for any b l ere a, b be bef befe bg ereg It implies l ere a ereg If 1 a is a left P P -element in R, then e a is a left P P -element in ere by using the similar method Thus ere is a left almost P P -ring An elementary argument using condition in Definition 21 shows that a direct product of rings is left P P if and only if each factor is left P P However, for left almost P P -rings we have the next result Theorem 26 Let R α I R α Then R is a left almost P P -ring if and only if there exists α 0 I, such that R α0 is a left almost P P -ring and for each α I α 0, R α is a left P P -ring Proof Let x x α R, α I By hypothesis, x α0 or 1 Rα0 x α0 is left P P If x α0 is a left P P -element in R α0, then x is a left P P -element in R If 1 Rα0 x α0 is a left P P -element in R α0, then 1 x is a left P P -element in R Thus, the result follows Assume that R is a left almost P P -ring Then every R α is also a left almost P P -ring Write R R α0 S, where S R α, α I α 0 If neither R α0 nor S is left P P, then we can find non-left P P -elements r R α0 and s S Choose a 1 Rα0 r, s Then neither a nor 1 a r, 1 S s is left P P in R, a contradiction Hence, either R α0 or S is a left P P -ring If S is a left P P -ring, the result follows If S is a left almost P P -ring, by iteration of this process, we complete the proof Remark 27 1 Note that the direct product of left almost P P -rings may not be a left almost P P -ring Clearly, Z 4 and Z 9 are almost P P -rings But we claim that Z 4 Z 9 is not an almost P P -ring Choose a 2, 4 Then neither a nor 1 a is P P in Z 4 Z 9, and we are done The

4 202 Yueming XIANG example also shows that the homomorphic image of a left almost P P -ring need not to be a left almost P P -ring 2 By the theorem above, if R S is a left almost P P -ring, then either R or S is left P P So, in general, the ring Z n of integers mod n is an almost P P -ring if and only if pq 2 does not divide n, where p and q are distinct primes It is easy to see that n 36 is the least positive integer such that Z n is not an almost P P -ring Let D be a ring and C a subring of D with 1 D C We set R[D, C] {d 1,, d n, c, c : d i D, c C, n 1} with addition and multiplication defined componentwise Since Nicholson used R[D, C] to construct rings which are semiregular but not regular, more and more algebraists use this structure to construct various counterexamples in ring theory Theorem 28 R[D, C] is a left almost P P -ring if and only if the following hold: 1 D is a left P P -ring 2 For any c C, there exists an e 2 e C such that l C c Ce, l D c De or l C 1 c Ce, l D 1 c De Proof For convenience, let S R[D, C] Assume that D is not a left P P -ring Then there exists a non-left P P -element x D Choose a x, 1 x, 1, 1, S By hypothesis, either a or 1 a is left P P in S If a is a left P P -element in S, then x is left P P in D, a contradiction If 1 a is left P P in S, then x is also left P P in D, a contradiction Thus, D is a left P P -ring To prove condition 2, let c C and c c, c, S Since S is a left almost P P - ring, either c or 1 c is left P P in S Assume that c is left P P, then l S c Se, where e e 1,, e m, e, e, and e i D, e C are also idempotents Thus Ce l C c and De l D c If x l C c, let x x, x, Then x l S c Se, and x a 1 e 1,, a m e m, be, be, Thus, by computing the m + 1th component of x, we have x be Ce, thus l C c Ce If s l D c, let s d 1, d 2,, d m+1, 0,, where d i s for i 1,, m + 1 Then s l S c Se, showing that s De, thus l D c De Assume that 1 c is left P P, then we have l C 1 c Ce, l D 1 c De by the similar argument Let a a 1,, a n, c, c, S For any x x 1,, x n,, x m, x, x, l S a, we have x i a i 0 i 1,, m, where a n+1 a m c, and xc 0 Note that l D a i De i i 1,, n If l C c Ce and l D c De, where e 2 e C are also idempotent So x i d i e i i 1,, n, x i d i e i n + 1,, m, x c e with all d i D, c C Thus x d 1,, d n, d n+1,, d m, c, c, e 1,, e n, e,, e, e, e, Se On the other hand, for any y y 1,, y m, y, y, Se, we have y i De i i 1,, n, y i De i n + 1,, m and y Ce Then y i a i 0 i 1,, n, y i c 0 i n + 1,, m and yc 0 It implies that y a 0, and hence y l S a Therefore, a is left P P in S

5 A generalization of VNL-rings and P P -rings 203 If l C 1 c Ce, l D 1 c De, using the similar argument above, we can prove 1 a is left P P in S Therefore, S is a left almost P P -ring By Theorem 28, we have the next corollaries immediately Corollary 29 R[D, D] is a left almost P P -ring if and only if D is a left P P -ring Corollary 210 R[D, C] is a left P P -ring if and only if D and C are left P P -rings and for any c C, there exists an e 2 e C such that l D c De Example 211 Let S R[D, C], where D Q and C Z Then S is an almost P P -ring by Theorem 28 But S is not a VNL-ring in view of the argument of [4, Example 25] Example 212 Let S R[D, C], where D M 2 Z 2 and C { a b 0 a a, b Z2 } Then S is an almost P P -ring which is not left P P, not local Proof Obviously, D M 2 Z 2 is a P P -ring C {, , , 0 1 } If c, let e ; If c or , let e 1 c , let e ; If c 0 1, consider In either case, we have l C c Ce, l D c De or l C 1 c Ce, l D 1 c De By Theorem 28, S is a left almost P P -ring Similarly, we can prove that S is a right almost P P -ring Choose c 0 1 C, Rc is not projective since lc c JC cannot be generated by an idempotent, then C is not a left P P -ring Thus S is not a left P P -ring by Corollary 210 Note that JS R[JD, JD JC] 0, then S is not local, otherwise, S is regular, a contradiction 3 Matrix extensions Matrix constructions will provide new sources of examples of left almost P P -rings this section, we will develop results which allows us to study when full matrices and triangular matrices are left almost P P -rings Lemma 31 Let R be a ring and a R Then the following are equivalent: 1 a R is a left P P -element 2 α a M2 R is a left P P -element 3 β a M2 R is a left P P -element Proof Write S M 2 R 1 2 If a R is left P P, there exists an idempotent e 2 e R such that l R a Re Hence e 0 a and e 0 2 e 0 ls α If b c m n ls α, b c a 0 ba c m n 0 1 ma n In

6 204 Yueming XIANG It implies that b, m l R a Re, c n 0 Then b r 1 e, m r 2 e, and so b c r 1 0 e 0 e 0 S m n r 2 0 We prove that l S α S e Assume that α a such that l S α SE So a 1 a 2 a 3 a 4 S is left P P, there exists an idempotent E a1 a 2 a 3 a 4 S a and hence a 2 a 4 0, a 1 a 2 1, a 3 a 1 a 3, a 1, a 3 l R a Conversely, if x l R a, then x 0 a 0, 0 1, and hence x 0 ls α SE Thus x 0 r 1 r 2 r 3 r 4 a 1 0 a 3 0 r 1 + r 2 a 3 a 1 0 r 3 a 1 + r 4 a 3 0 It implies that x r 1 + r 2 a 3 a 1 Ra 1 Therefore, l R a Ra 1, where a 2 1 a 1 R 1 3 is similar to the proof of 1 2 Now we are in a position to prove when a matrix ring is a left almost P P -ring The following result is a generalization of [16, Proposition 763] Theorem 32 Let R be a ring Then the following are equivalent: 1 R is a left semihereditary ring; 2 M n R is a left P P -ring for every n 1; 3 M n R is a left almost P P -ring for every n 1 Proof 1 2 is dual to [16, Proposition 763] 2 3 is trivial 3 2 It is enough to show that if M 2 R is left almost P P, then R is left P P For any a R Choose A a a a 1 a M2 R By hypothesis, either A M 2 R or I 2 A M 2 R is left P P Suppose that A M 2 R is left P P Note that a A, 0 1 a by Remark 223, a M2 R is left P P So a R is left P P by Lemma 31 If I 2 A M 2 R is left P P, noting that a 1 0 I 2 A, 0 a by Remark 223, we have 0 a 1 0 M2 R is left P P So a R is left P P by Lemma 31 again

7 A generalization of VNL-rings and P P -rings 205 By the theorem above and Example 23, being a left almost P P -ring is not Morita invariant The next example shows that the definition of almost P P -rings is not left-right symmetric Example 33 Let S be a von Neumann regular ring with an ideal I such that, as a submodule of S, I is not a direct summand Let R S/I and T R R 0 S By the augment of [16, Example 234], T is left semihereditary but not right semihereditary Then there exists some n n 2 such that the matrix ring M n T is a left almost P P ring but not right almost P P A ring R is said to be right Kasch if every simple right R-module embeds in R R Proposition 34 If R is a right Kasch and left almost P P ring, then it is a right almost P P - ring Proof For any a R, a or 1 a is left P P in R Assume that a is a left P P -element in R There exists e 2 e R such that la R1 e Then a ea, and hence ar er Now we prove that ar er Otherwise, ar M, where M is a maximal submodule of er Since R is right Kasch, there exists a monomorphism f : er/m R by fe + M b Then eb b and ba 0 So b la R1 e, and hence b be 0 Since f is a monomorphism, e M, contradicting with the maximality of M So ar er is projective It implies that a is a right P P -element Assume that 1 a is a left P P -element We can prove 1 a is a right P P -element by the similar method Let R and S be rings and R M S a bimodule We write the generalized triangular matrix as T R M 0 S Following [13], a left module is a P P -module if every principal submodule is projective Now we consider the necessary and sufficient conditions of what a generalized triangular matrix ring is left almost P P Proposition 35 Let R and S be rings and R M S a bimodule If the following hold: 1 R is left P P and S is left almost P P ; 2 If b S is a left P P -element, then l M b Ml S b and M/Mb is a left P P -module If b S is not a left P P -element, then l M 1 b Ml S 1 b and M/M1 b is a left P P -module then T R M 0 S is a left almost P P -ring Proof For any α a m 0 b T If b is left P P in S, then ls b Sf with f 2 f S Note that l R a Re 1, l R m + Mb Re 2, where e 2 i e i R, i 1, 2 By [13, Lemma 1], Re 1 Re 2 Re for e 2 e R So e Re 1, Re 2, we can let em m 1 b for some m 1 M Write m 2 em 1 1 f, then m 2 b em, and hence β e m 2 0 f lt α Conversely, if x y 0 s lt α, xa 0, sb 0, xm yb, and so xe x, sf s and y + xm 2 b yb + xm 2 b xm + xem xm + xm 0 Hence y + xm 2 l M b Ml S b, and we have y + xm 2 y + xm 2 f It follows that x y 0 s x y+xm2 e m2 0 s 0 f T β Thus α is a left P P -element in T If b S is not left P P -element, then 1 b is left P P because S is a left almost P P -ring Using the similar method above, we can prove that 1 α 1 a m 0 1 b is a left P P -element in T Therefore, T is left almost P P

8 206 Yueming XIANG Proposition 36 Let R and S be rings and R M S a bimodule If T R M 0 S is a left almost P P -ring, then one of R and S is left P P and the other is left almost P P Proof The result follows from Proposition 252 and Lemma 41 below Corollary 37 Let T n R be the rings of upper triangular matrices over R Then the following are equivalent: 1 R is regular; 2 T n R is a left P P -ring for every n 1; 3 T n R is a left almost P P -ring for every n 1 Proof It follows by [13, Theorem 4] and Proposition 36 4 Semiperfect left almost P P -rings Now we consider the structure of semiperfect, left almost P P -rings Lemma 41 If R is a left almost P P -ring and e 2 e R, then either ere or 1 er1 e is a left P P -ring Proof We have the Pierce decomposition R ere er1 e 1 ere 1 er1 e If x ere and y 1 er1 e are not left P P -elements, then neither a x y nor 1 a 1 x 0 0 y are left P P -elements Recall a ring R is abelian if each idempotent in R is central An element a of a ring R is called an exchange element if there exists an idempotent e R such that e Ra and 1 e R1 a The ring R is an exchange ring if and only if every element of R is an exchange element Proposition 42 The following are equivalent for an abelian, exchange ring R 1 R is an almost P P -ring; 2 For every e 2 e R, either ere or 1 er1 e is a left P P -ring Proof 1 2 It follows by Lemma For any a R, as R is an exchange ring, there exists e 2 e R such that e Ra and 1 e R1 a So Ra + R1 e R and R1 a + Re R It implies that Rae Re and R1 a1 e R1 e Thus ae is left P P -element in Re and 1 a1 e is left P P -element in R1 e Now if ere Re is left P P, then 1 ae is a left P P -element in ere, and hence 1 a 1 ae + 1 a1 e is left P P in R Similarly, if 1 er1 e R1 e is left P P, then a is left P P in R Therefore, R is an almost P P -ring Lemma 43 Let R be a local ring Then R is a left P P -ring if and only if R is a domain

9 A generalization of VNL-rings and P P -rings 207 Proposition 44 Let R be a semiperfect, left almost P P -ring with 1 e 1 + e 2, where e 1, e 2 are orthogonal local idempotents Then R is isomorphic to one of the following: 1 M 2 D for some domain D; 2 D 1 Y X D 2, where D1 is a domain, D 2 is a local ring and XY JD 1, Y X JD 2 In particular, if R is also abelian, then R M 2 D or R A B, where D, A are domains and B is a local ring Proof We use the Pierce decomposition R e 1 Re 1 e 1 Re 2 e 2 Re 1 e 2 Re 2 If e 1 R e 2 R, then R M 2 e 1 Re 1, where e 1 Re 1 is a local left P P -ring by Lemma 41 So e 1 Re 1 is a domain If e 1 R e 2 R, then e 1 Re 2 JR and e 2 Re 1 JR by [4, Lemma 42] We assume that e 1 Re 1 is a local left P P -ring by Lemma 41, and hence e 1 Re 1 is a domain Note e 1 Re 2 Re 1 e 1 Re 1 JR Je 1 Re 1 and e 2 Re 1 Re 2 e 2 Re 2 JR Je 2 Re 2 So write D 1 e 1 Re 1, D 2 e 2 Re 2, X e 1 Re 2 and Y e 2 Re 1, then 2 follows Proposition 45 Let R be a semiperfect, left almost P P -ring with 1 e 1 + e 2 + e 3, where e 1, e 2, e 3 are orthogonal local idempotents Then R is isomorphic to one of the following: 1 M 3 D for some domain D; 2 D 1 Y X D 2, where D1 is a domain, D 2 is a local ring and XY JD 2, Y X JD 1 ; 3 D 1 Y X D 2, where D1 is a prime ring, D 2 is a local ring and XY JD 2, Y X JD 1 ; 4 S Y X D with S D1 Y 1 X 1 D 2 and D1, D 2, D are domains, X 1 Y 1 JD 2, Y 1 X 1 JD 1, XY JD, Y X JS Proof Case 1 If e i R e j R for i, j 1, 2, 3, then R M 3 e 1 Re 1, where e 1 Re 1 is a local left P P -ring by Lemma 41 So e 1 Re 1 is a domain We now consider the the Pierce decomposition R 1 e 1 R1 e 1 1 e 1 Re 1 e 1 R1 e 1 e 1 Re 1 Case 2 Assume that e 1 Re 1 is local but not a left P P -ring by Lemma 41, then 1 e 1 R1 e 1 is a domain, and hence e 2 Re 2 and e 3 Re 3 are also domains By [4, Lemma 42], e 1 Re 2, e 2 Re 1, e 1 Re 3 and e 3 Re 1 are all contained in JR So 1 e 1 Re 1 R1 e 1 JR 1 e 1 R1 e 1 J1 e 1 R1 e 1 and e 1 R1 e 1 Re 1 JR e 1 Re 1 Je 1 Re 1 Thus R is isomorphic to the ring in 2 Case 3 Assume that e i Re i is a domain for i 1, 2, 3 If e 1 R e 2 R but e 2 R e 3 R, then 1 e 1 R1 e 1 M 2 D for some domain D, and hence 1 e 1 R1 e 1 is a prime ring By [4, Lemma 42], e 1 Re 2, e 2 Re 1, e 1 Re 3 and e 3 Re 1 are all contained in JR So 1 e 1 Re 1 R1 e 1 JR 1 e 1 R1 e 1 J1 e 1 R1 e 1 and e 1 R1 e 1 Re 1 JR e 1 Re 1 Je 1 Re 1 Then 3 is done

10 208 Yueming XIANG Case 4 Assume that e i Re i is a domain for i 1, 2, 3 and e 1 R e 2 R e 3 R Then 1 e 1 R1 e 1 e 2 Re 2 e 2 Re 3, e 3 Re 2 e 3 Re 3 where e 2 Re 3 Re 2 Je 2 Re 2 and e 3 Re 2 Re 3 Je 3 Re 3 Note that 1 e 1 Re 1 R1 e 1 JR 1 e 1 R1 e 1 J1 e 1 R1 e 1 So write e 2 Re 2 D 1, e 3 Re 3 D 2, e 3 Re 2 X 1, e 2 Re 3 Y 1, e 1 Re 1 D, 1 e 1 Re 1 X, e 1 R1 e 1 Y, then 4 is also done Acknowledgements leading to the improvement of the paper References The authors are indebted to the referees for their valuable comments [1] M CONTESSA On certain classes of pm-rings Comm Algebra, 1984, : [2] E A OSBA, M HENRIKSEN, O ALKAM Combining local and von Neumann regular rings Comm Algebra, 2004, 327: [3] Weixing CHEN, Wenting TONG On noncommutative VNL-rings and GVNL-rings Glasg Math J, 2006, 481: [4] H K GROVER, D KHURANA Some characterizations of VNL rings Comm Algebra, 2009, 379: [5] Y HIRANO, M HONGAN, M ÔHORI On right pp rings Math J Okayama Univ, 1982, 242: [6] C HUH, H K KIM, Y LEE pp rings and generalized pp rings J Pure Appl Algebra, 2002, 1671: [7] I KAPLANSKY Rings of Operators Benjamin, New York, 1965 [8] Y LEE, C HUH Counterexamples on pp-rings Kyungpook Math J, 1998, 382: [9] Zhongkui LIU, Renyu ZHAO A generalization of PP-rings and pq-baer rings Glasg Math J, 2006, 482: [10] Lixin MAO, Nanqing DING, Wenting TONG New characterizations and generalizations of PP rings Vietnam J Math, 2005, 331: [11] Lixin MAO Generalized P -flatness and P -injectivity of modules Hacet J Math Stat, 2011, 401: [12] W K NICHOLSON, J F WATTERS Rings with projective socle Proc Amer Math Soc, 1988, 1023: [13] W K NICHOLSON On PP-rings Period Math Hungar, 1993, 272: [14] M ÔHORI On non-commutative generalized pp rings Math J Okayama Univ, 1984, 26: [15] Zhong YI, Yiqiang ZHOU Baer and quasi-baer properties of group rings J Aust Math Soc, 2007, 832: [16] T Y LAM Lectures on Modules and Rings Springer-Verlag, New York, 1999