Small Principally Quasi-injective Modules

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1 Int. J. Contemp. Math. Sciences, Vol. 6, 2011, no. 11, Small Principally Quasi-injective Modules S. Wongwai Department of Mathematics Faculty of Science and Technology Rajamangala University of Technology Thanyaburi Pathumthani 12110, Thailand Abstract Let M be a right R-module. A right R-module N is called small principally M-injective (briefly, SP-M-injective) if, every R-homomorphism from a small and principal submodule of M to N can be extended to an R-homomorphism from M to N. In this paper we give some characterizations and properties of small principally quasi-injective modules. Mathematics Subject Classification: 16D50, 16D70, 16D80 Keywords: SP-injective rings, SPQ-injective Modules and Endomorphism Rings 1. Introduction Throughout this paper, R will be an associative ring with identity and all modules are unitary right R-modules. For right R-modules M and N, Hom R (M,N) denotes the set of all R-homomorphisms from M to N and S = End R (M) denotes the endomorphism ring of M. If X is a subset of M the right (resp. left) annihilator of X in R (resp. S) is denoted by r R (X) (resp. l S (X)). By notations, N M, N e M, and N M we mean that N is a direct summand, an essential submodule and a superfluous submodule of M, respectively. We denote the Jacobson radical of M by J(M). Let R be a ring. A right R-module M is called principally injective (or P -injective), if every R-homomorphism from a principal right ideal of R to M can be extended to an R-homomorphism from R to M. Equivalently, l M r R (a) = Ma for all a R. This notion was introduced by Camillo [2] for commutative rings. In [6], Nicholson and Yousif studied the structure of principally injective rings and gave some applications. Nicholson, Park, and Yousif [8] extended this notion of principally injective rings to the one for

2 528 S. Wongwai modules. In [11], L.V. Thuyet, and T.C. Quynh, introduced the definition of small principally injective modules, a right R- module M is called small principally injective (or SP injective) if, every R-homomorphism from a small and principal right ideal ar to M can be extended to an R-homomorphism from R to M. A ring R is called right SP-injective if R R is SP-injective. In this note we introduce the definition of small principally quasi-injective modules and give some characterizations and properties. Some results on principally quasi-injective modules [8] are extended to these modules. 2. SPQ-injective Modules Recall that a submodule K of a right R-module M is superfluous (or small) in M, abbreviated K M, in case for every submodule L of M, K + L = M implies L = M. Definition 2.1. Let M be a right R-module. A right R-module N is called small principally M-injective (briefly, SP-M-injective) if, every R-homomorphism from a small and principal submodule of M to N can be extended to an R- homomorphism from M to N. Lemma 2.2. Let M and N be right R-modules. Then N is SP-M-injective if and only if each m M with mr M, Hom R (M,N)m = l N r R (m). Proof. Clearly, Hom R (M,N)m l N r R (m). Let x l N r R (m). Define ϕ : mr xr by ϕ(mr) =xr for every r R. Then ϕ is well-defined because r R (m) r R (x). It is clear that ϕ is an R-homomorphism. Since N is SP-Minjective, there exists an R-homomorphism ϕ : M N such that ϕι 1 = ι 2 ϕ, where ι 1 : mr M and ι 2 : xr N are the inclusion maps. Hence x = ϕ(m) = ϕ(m) Hom R (M,N)m. Conversely, let m M with mr M, and let ϕ : mr N be an R- homomorphism. Then ϕ(m) l N r R (m) so by assumption, ϕ(m) = ϕ(m) for some ϕ Hom R (M,N). This show that N is SP-M-injective. Example 2.3. Let R = ( ) F F 0 F where F is a field, MR = R R and N R = ( ) F F 0 0. Then N is SP-M-injective. Proof. It is clear that only X = ( ) 0 F 0 0 is the nonzero small and principal submodule of M. Let ϕ : X N be R-homomorphism. Since ( ) X, there exists x 1,x 12 F such that ϕ( ( ) ( ) ( ) ( )( ) )= x11 x Then ϕ( )=ϕ[ ] = ϕ( ( ) ( ) ( )( ) ( ) ) = x11 x = 0 x It follows that x11 = 0. Define ϕ : M N by ϕ( ( ) ( ) )= x It is clear that ϕ is an R-homomorphism.

3 Small principally quasi-injective modules 529 Then ϕ( ( ) ( )( ) ( ) ( ) ( ) )= ϕ[ ]= ϕ( x ) = 0 x This show that ϕ is an extension of ϕ. Thus N is SP-M-injective. Lemma 2.4. Let N i (1 i n) be SP-M-injective modules. Then n i=1 N i is SP-M-injective. Proof. It is enough to prove the result for n =2. Let m M with mr M, and ϕ : mr N 1 N2 be an R-homomorphism. Since N 1 and N 2 are SP- M-injective, there exists R-homomorphisms ϕ 1 : M N 1 and ϕ 2 : M N 2 such that ϕ 1 ι = π 1 ϕ and ϕ 2 ι = π 2 ϕ where π 1 and π 2 are the projection maps from N 1 N2 to N 1 and N 2, respectively, and ι : mr M is the inclusion map. Put ϕ = ι 1 ϕ 1 + ι 2 ϕ 2 : M N 1 N2. Thus it is clear that ϕ extends ϕ. Lemma 2.5. Any direct summand of an SP-M-injective module is again SP-M-injective. Proof. By definition. Theorem 2.6. The following conditions are equivalent for a projective module M: (1) Every small and principal submodule of M is projective. (2) Every factor module of an SP-M-injective module is SP-M-injective. (3) Every factor module of an injective R-module is SP-M-injective. Proof. (1) (2) Let N be an SP-M-injective module, X a submodule of N, mr M, and let ϕ : mr N/X be an R-homomorphism. Then by (1), there exists an R-homomorphism α : mr N such that ϕ = ηα where η : N N/X is natural R-epimorphism. Hence α can be extended to an R-homomorphism β : M N. Then ηβ is an extension of ϕ to M. (2) (3) is clear. (3) (1) Let mr M, α : A B be an R-epimorphism, and let ϕ : mr B be an R-homomorphism. Embed A in an injective module E [1, 18.6]. Then B A/Ker(α) is a submodule of E/Ker(α) so by hypothesis, ϕ can be extended to ϕ : M E/Ker(α). Since M is projective, ϕ can be lifted to β : M E. It is clear that β(mr) A. Therefore we have lifted ϕ. 3. The Endomorphism Ring A right R-module M is called small principally quasi-injective (briefly, SP Q-injective) ifitissp-m-injective. Lemma 3.1. Let M be a right R module and S = End R (M). Then the following conditions are equivalent:

4 530 S. Wongwai (1) M is SPQ injective. (2) l M r R (m) =Sm for all m M with mr M. (3) r R (m) r R (n), where m, n M with mr M, implies Sn Sm. (4) l M (r R (m) ar) =l M (a)+sm for all a R and m M with mr M. Proof. (1) (2) by Lemma 2.2. (2) (3) If r R (m) r R (n), where m, n M with mr M, then l M r R (n) l M r R (m). Since Sn l M r R (n) and by (2), l M r R (m) =Sm, so we have Sn Sm. (3) (4) Let a R, m M with mr M, and let x l M (r R (m) ar). Then r R (ma) r R (xa) sosxa Sma by (3) because mar M. Thus xa = ϕ(ma), ϕ S and hence (x ϕ(m)) l M (a). It follows that x l M (a)+sm. The other inclusion is clear. (4) (2) Put a =1. Lemma 3.2. Let M be an SPQ-injective module and S = End R (M). If m M and α S with α(m) M, then l S (Ker(α) mr) =l S (m)+sα. Proof. It is always the case that l S (m) +Sα l S (Ker(α) mr). Let β l S (Ker(α) mr). Then r R (α(m)) r R (β(m)), so l M r R (β(m)) l M r R (α(m)). Since α(m)r M, Sβ(m) l M r R (β(m)) l M r R (α(m)) = Sα(m) by Lemma 3.1, so β(m) =γα(m), γ S. It follows that (β γα) l S (m), and hence β l S (m)+sα. Following [8], a right R-module M is called a principal self-generator if every element m M has the form m = γ(m 1 ) for some γ : M mr. Proposition 3.3. Let M be a principal module which is a principal selfgenerator and let S = End R (M). Then the following conditions are equivalent: (1) M is SPQ injective. (2) l S (Ker(α) mr) =l S (m)+sα for all m M and α S with α(m) M. (3) l S (Ker(α)) = Sα for all α S with α(m) M. (4) Ker(α) Ker(β), where α, β S with α(m) M, implies Sβ Sα.

5 Small principally quasi-injective modules 531 Proof. (1) (2) by Lemma 3.2. (2) (3) If M = m 0 R, take m = m 0 in (2). (3) (4) If Ker(α) Ker(β), then l S (Ker(β)) l S (Ker(α). It follows that Sβ l S (Ker(β)) l S (Ker(α) =Sα. (4) (1) Let m M with mr M and ϕ : mr M be an R- homomorphism. Since M is a principal self-generator, there exists β S such that β(m 1 )=m, so Ker(β) Ker(ϕβ) and β(m) M. Then by (4), Sϕβ Sβ, write ϕβ = ϕβ, ϕ S. This shows that ϕ extends ϕ. Theorem 3.4. Let M be an SPQ injective module, m, n M and mr M. (1) If mr embeds in nr, then Sm is an image of Sn. (2) If nr is an image of mr, then Sn embeds in Sm. (3) If mr nr, then Sm Sn. Proof. (1) Let f : mr nr be an R monomorphism. Let ι 1 : mr M and ι 2 : nr M be the inclusion maps. Since M is SPQ-injective, there exists an R homomorphism f : M M such that ι 2 f = fι 1. Let σ : Sn Sm defined by σ(α(n)) = α f(m) for every α S. Since σ(α(n)) = αf(m) α(nr), σis well-defined. It is clear that σ is an S homomorphism. Note that f(m)r = f(m)r M by [1, Lemma 5.18]. Since f is monic, r R (f(m)) r R (m) and hence by Lemma 3.1, Sm Sf(m). Then m Sf(m) σ(sn). (2) By the same notations as in (1), let f : mr nr be an R epimorphism. Write f(ms) =n, s R. Since M is SPQ-injective, f can be extended to f : M M such that ι 2 f = fι 1. Define σ : Sn Sm by σ(α(n)) = α f(ms) for every α S. It is clear that σ is S-homomorphism. If α(n) Ker(σ), then 0 = σ(α(n)) = α f(ms) = αf(ms) = α(n). This show that σ is an S monomorphism. (3) Follows from (1) and (2). Lemma 3.5. Let M be a principal module which is a self generator. If M is SPQ-injective, then S is a right SP-injective ring. Proof. Let s J(S) and ϕ : ss S be an S-homomorphism. Since M is a self generator Ker(s) = t I t(m) for some I S. Write ϕ(s) =g where g S. For any t I we have gt = ϕ(st) = 0 and hence Im(t) Ker(g). It follows that Ker(s) Ker(g). Then there exists an R homomorphism α : s(m) M such that αs = g. Since M is a principal module, J(M) M so we have J(S)M J(M), it follows that s(m) is a small and principal submodule of M. Then by assumption, α can be extended to an R homomorphism α : M M such that αι = α where ι : s(m) M is the inclusion map. Hence αιs = αs = g. Define ϕ : S S by ϕ(f) = αf for every f S. It is clear that ϕ is an S homomorphism. Then ϕ(sf) = αsf = gf = ϕ(sf). Therefore S is a right SP injective ring.

6 532 S. Wongwai Following [7], A ring R is called right mininjective if every isomorphism between simple right ideals is given by left multiplication by an element of R. Equivalently, if kr is simple, k R, lr(k) =Rk. Proposition 3.6. Let M be a principal module which is a self generator. If M is SPQ-injective, then (1) S is right mininjective. (2) lr(ss)=ss. for any s J(S). (3) If ss ts and Ss St are both direct, s, t J(S), then l(s)+l(t) =S. Proof. (1) We have S is SP-injective by Lemma 3.5. Since every simple right ideal of S is either nilpotent or a direct summand of S [4, (10.22) Brauer s Lemma], each right SP-injective ring is right mininjective ring. (2) Let α lr(ss). Define ϕ : ss αs by ϕ(su) =αu for every u S. Since su = 0 implies αu = 0,ϕis well-defined. It is clear that ϕ is an S-homomorphism. Since S is right SP-injective, there exists an extension ϕ : S S of ϕ. Hence α = ϕ(s) = ϕ(1)s Ss. This shows that lr(ss) Ss. The inclusion Ss lr(ss) is always holds. (3) Define ϕ :(s+t)s S by ϕ(s+t)u = tu for every u S. If (s+t)u = 0, then su = tu ss ts =0sotu =0. This shows that ϕ is well-defined. It is clear that ϕ is an S homomorphism. Since S is right SP-injective, there exists an extension ϕ : S S of ϕ. Hence ϕ(1)(s + t) =ϕ(s + t) =t so ϕ(1)s =(1 ϕ(1))t Ss St =0. Then ϕ(1) l(s) and (1 ϕ(1)) l(t). Hence 1 = ϕ(1) + (1 ϕ(1)) l(s)+l(t). It follows that l(s)+l(t) =S. Proposition 3.7. Let M be an SPQ-injective module and m i M with m i R M, (1 i n). (1) If Sm 1... Sm n is direct, then any R-homomorphism α : m 1 R m n R M has an extension in S. (2) If m 1 R... m n R is direct, then S(m m n ) = Sm Sm n. Proof. (1) Since M is SPQ-injective, for each i, there exists an R-homomorphism ϕ i : M M such that ϕ i (m i ) = α(m i ). Since ( n i=1 m i )R M and ( n i=1 m i )R n i=1 m i R, α can be extended to ϕ : M M such that, for any r R, n n ϕ( m i )r = α( m i )r. i=1 It follows that n i=1 ϕ(m i )= n i=1 ϕ i (m i ). Since Sm 1... Sm n is direct, ϕ(m i )=ϕ i (m i ) for all (1 i n). Therefore ϕ is an extension of α. i=1

7 Small principally quasi-injective modules 533 (2) Let α 1 (m 1 ) α n (m n ) Sm Sm n. For each i, define ϕ i : (m m n )R M by ϕ i ((m m n )r) = m i r for every r R. Since m 1 R... m n R is direct, ϕ i is well-defined, so it is clear that ϕ i is an R-homomorphism. By assumption and (m m n )R M, there exists ϕ i S which is an extension of ϕ i. Then m i = ϕ i (m m n )= ϕ i (m m n ) S(m m n ). Consequently, α 1 (m 1 ) α n (m n ) S(m m n ). This shows that Sm Sm n S(m m n ). The other inclusion always holds. Proposition 3.8. Let M be an SPQ injective module, A a fully invariant submodule of M, and let B 1,..., B n be small and fully invariant submodules of M. If B 1... B n is direct, then A (B 1... B n )=(A B 1 )... (A B n ). Proof. Always (A B 1 )... (A B n ) A (B 1... B n ). Let a = n i=1 b i A (B 1... B n ) and let π k : n i=1b i R b k R be the projection map. Since, for each i, B i is small and fully invariant, Sb i B i so n i=1 Sb i is direct. Then by Proposition 3.7, π k has an extension π k in S. It follows that b i = π i (a) A B i Therefore a n i=1(a B i ). References [1] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Graduate Texts in Math.No.13,Springer-verlag, New York, [2] V. Camillo, Commutative rings whose principal ideals are annihilators, Portugal Math., 46(1989), [3] N. V. Dung, D. V. Huynh, P. F. Smith and R. Wisbauer, Extending Modules, Pitman, London, [4] T.Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics Vol 131, Springer-Verlag, New York, [5] S. H. Mohamed and B. J. Müller, Continuous and Discrete Modules, London Math. Soc. Lecture Note Series 14, Cambridge Univ. Press, [6] W. K. Nicholson and M. F. Yousif, Principally injective rings, J. Algebra, 174(1995), [7] W. K. Nicholson and M. F. Yousif, Mininjective rings, J. Algebra, 187(1997),

8 534 S. Wongwai [8] W. K. Nicholson, J. K. Park and M. F. Yousif, Principally quasi-injective modules, Comm. Algebra, 27:4(1999), [9] N. V. Sanh, K. P. Shum, S. Dhompongsa and S. Wongwai, On quasiprincipally injective modules, Algebra Coll.6: 3(1999), [10] L. Shen and J. Shen Small injective rings, arxiv: Math., RA/ v.1 (2005). [11] L.V. Thuyet, and T.C.Quynh, On small injective rings, simple-injective and quasi-frobenius rings, Acta Math. Univ. Comenianae, Vol.78(2), (2009) pp [12] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach London, Tokyo e.a., [13] S. Wongwai, On the endomorphism ring of a semi-injective module, Acta Math. Univ. Comenianae, Vol.71,1(2002), pp Received: September, 2010