ISSN 2148-838Xhttp://dx.doi.org/10.13069/jacodesmath.284943 J. Algebra Comb. Discrete Appl. 4(2) • 123–129 Received: 17 June 2015 Accepted: 5 February 2016 Journal of Algebra Combinatorics Discrete Structures and Applications On some radicals and proper classes associated to simple modules Research Article Septimiu Crivei, Derya Keskin Tütüncü Abstract: For a unitary right module M, there are two known partitions of simple modules in the category σ[M]: the first one divides them into M-injective modules and M-small modules, while the second one divides them into M-projective modules and M-singular modules. We study inclusions between the first two and the last two classes of simple modules in terms of some associated radicals and proper classes. 2010 MSC: 16S90, 16D60, 16D50, 16D40, 16D90 Keywords: Radical, Proper class, Simple module, Relative supplement submodule, M-injective module, M-projective module, M-small module, M-singular module 1. Introduction For an associative ring R with identity, let M be a unitary right R-module. Then the category σ[M] is a Grothendieck category which consists of all right R-modules subgenerated by M, that is, submodules of M-generated right R-modules. An object X of σ[M] is called: 1. M-injective if the functor HomR(−,X) preserves exactness of all short exact sequences 0 → K → M → N → 0. 2. M-projective if the functor HomR(X,−) preserves exactness of all short exact sequences 0 → K → M → N → 0. 3. M-small if X is superfluous in some module Y ∈ σ[M]. 4. M-singular if X ∼= Y/Z for some essential submodule Z of a module Y ∈ σ[M]. Septimiu Crivei; Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania (email: crivei@math.ubbcluj.ro). Derya Keskin Tütüncü (Corresponding Author); Department of Mathematics, Hacettepe University, 06800, Ankara, Turkey (email: keskin@hacettepe.edu.tr). 123 S. Crivei, D. Keskin Tütüncü / J. Algebra Comb. Discrete Appl. 4(2) (2017) 123–129 There are two classical dichotomies concerning the simple modules in σ[M]. The first one partitions them into the pair of classes (S(I ), S(M )), where S(I ) is the class of simple M-injective modules in σ[M] and S(M ) is the class of simple M-small modules in σ[M] [4, 4.2]. The second one partitions them into the pair of classes (S(P), S(S )), where S(P) is the class of simple M-projective modules in σ[M] and S(S ) is the class of simple M-singular modules in σ[M] [3, 8.2]. In the present note we give equivalent conditions under which the above two pairs of classes of simple modules may be linked in terms of some associated radicals and proper classes. More precisely, we are interested in such characterizations for possible inclusions between the four classes of simple modules S(I ), S(M ), S(P) and S(S ). They are inspired by and complete results by Preisser Montaño [5]. 2. Preliminaries In this paper R will be an associative ring with identity and any module M will be a unitary right R-module. 2.1. Radicals Let A be an abelian category. We first recall the definitions of preradicals and radicals from [6, Chapter 6, §1]. A preradical τ of A is a subfunctor of the identity functor on A, that is, a functor τ : A−→A such that: 1. τ(A) ⊆ A for every object A of A. 2. τ(f) = f |τ(A): τ(A) −→ τ(B) for every morphism f : A −→ B in A. A preradical τ of A is idempotent if ττ = τ. Preradicals of A are partially ordered by the relation defined by τ ≤ τ′ if and only if τ(A) ⊆ τ′(A) for every object A of A. If the abelian category A is complete and cocomplete, then τ(⊕i∈IAi) = ⊕i∈Iτ(Ai) and τ( ∏ i∈I Ai) ⊆ ∏ i∈I τ(Ai) for any family {Ai}i∈I of objects of A and any preradical τ of A. Note that τ(P) = Pτ(RR) for all projective modules P in the category Mod-R of right R-modules and any preradical τ of Mod-R. A preradical τ of A is called radical if τ(A/τ(A)) = 0, for all objects A of A. In particular, one has the Jacobson radical Rad on a Grothendieck category, where Rad(X) is the intersection of maximal subobjects of X for every object X. Let τ be a radical of σ[M]. Let L be a submodule of a module N ∈ σ[M]. Then L is called a τ- supplement in N if there exists a submodule X of N such that N = L+X and L∩X ⊆ τ(L), equivalently, N = L + X and L∩X = τ(L) (see [1, 1.11] or [5, 4.9]). A similar concept may also be given with respect to a class C of modules in σ[M]. A submodule K of a module N ∈ σ[M] is called C-small in N if for every submodule X ≤ N, the equality K + X = N and N/X ∈ C imply X = N. We denote this by K �C N (see [5, 8.1]). For the class S of M-singular modules, note that S -small submodules are the δ-small submodules in σ[M] as defined by Zhou in [7] (see [5, 8.2(i)]). A submodule L of a module N ∈ σ[M] is called a C-supplement in N if there exists a submodule L′ of N such that L + L′ = N and L∩L′ �C L (see [5, 9.1]). Denote by S(C) the class of simple modules in C, and let N ∈ σ[M]. Then denote radS(C)(N) = Rej(N,S(C)) = ⋂ {Ker(f) | f : N −→ S,S ∈S(C)} (see [5, 10.1]). Note that radS(C) is a radical, Rad ≤ radS(C) and radS(C)(N) = N if and only if N has no nonzero simple factor modules in C (see [5, 10.2]). On the other hand, if C is closed under submodules and factor modules, then we also have radS(C)(N) = ∑ {L ≤ N | L �C N} 124 S. Crivei, D. Keskin Tütüncü / J. Algebra Comb. Discrete Appl. 4(2) (2017) 123–129 (see [5, 10.3]). Then radS(C)(N) = N if and only if every finitely generated submodule of N is C-small in N (see [5, 10.2 and 10.3]). 2.2. Proper classes Let A be an abelian category. We now recall the definition of proper classes from [2] (also see [5, Chapter 2, Section 3]). Let P be a class of short exact sequences in A. If 0 −→ A f−→ B g−→ C −→ 0 belongs to P, then f is called a P-monomorphism and g a P-epimorphism. A class P of short exact sequences in A is called a proper class if it satisfies the following axioms: (P1) P is closed under isomorphisms. (P2) P contains all splitting short exact sequences of A. (P3) If f and f′ are P-monomorphisms and the composition f′f is defined, then f′f is a P- monomorphism. (P4) If f and f′ are monomorphisms such that the composition f′f is defined and is a P-monomorphism, then f is a P-monomorphism. (P5) If g and g′ are P-epimorphisms and the composition gg′ is defined, then gg′ is a P-epimorphism. (P6) If g and g′ are epimorphisms such that the composition gg′ is defined and is a P-epimorphism, then g is a P-epimorphism. Proper classes in A are partially ordered by the relation defined by P ⊆P′ if and only if every short exact sequence in P belongs to P′. The class of all short exact sequences in σ[M] 0 −→ A −→ B −→ C −→ 0 such that A is a C-supplement in B, is a proper class and it is denoted by C-Suppl (see [5, 9.2]). The class of all short exact sequences in σ[M] 0 −→ A −→ B −→ C −→ 0 such that A is a radS(C)-supplement in B, is a proper class and it is denoted by radS(C)-Suppl (see [5, 10.4]). If C = σ[M], then radS(C) = Rad. And if C is a class in σ[M] closed under submodules and factor modules, then C-Suppl ⊆ radS(C)-Suppl (see [5, 10.7]). 3. Class inclusions Following [5, 10.20] and keeping the notation from the introduction, we consider the radicals α = radS(I ), β = radS(P), γ = radS(M ), 125 S. Crivei, D. Keskin Tütüncü / J. Algebra Comb. Discrete Appl. 4(2) (2017) 123–129 δ = radS(S ) of σ[M] associated to the corresponding classes of simple modules which are M-injective, M-projective, M-small and M-singular respectively. Note that Rad = α ∩ γ = β ∩ δ and α is an idempotent radical (see [5, 10.22]). Then we also have the proper classes α-Suppl, β-Suppl, γ-Suppl and δ-Suppl associated to the radicals α, β, γ and δ. We begin with four class inclusions for which we have equivalent conditions in terms of both associated radicals and associated proper classes. We first recall the following result, which motivated our study. Proposition 3.1. [5, Proposition 10.26] Let M be a module. The following conditions are equivalent: 1. S(S ) ⊆S(I ), i.e. every M-singular simple module is M-injective. 2. α ≤ δ. 3. α-Suppl ⊆ δ-Suppl. Proposition 3.2. Let M be a module. The following are equivalent: 1. S(M ) ⊆S(P), i.e. every M-small simple module is M-projective. 2. β ≤ γ. 3. β-Suppl ⊆ γ-Suppl. Proof. We only need to prove (3) ⇒ (1). Let S be an M-small simple module. Then γ(S) = 0. Since S is simple, we have β(S) = 0 or β(S) = S. Assume that β(S) = S. Then the short exact sequence 0 −→ S −→ EM(S) −→ EM(S)/S −→ 0 belongs to β-Suppl, and hence it belongs to γ-Suppl by hypothesis, where EM(S) is the M-injective hull of S in σ[M]. So, there exists a submodule T of EM(S) such that EM(S) = S +T and S∩T = γ(S) = 0. This means that S is M-injective, a contradiction. Therefore β(S) = 0. Now there exists a nonzero homomorphism f : S −→ S ′ with S ′ M-projective simple. Thus S is M-projective. One obtains the following two propositions in a similar way. Proposition 3.3. Let M be a module. The following are equivalent: 1. S(P) ⊆S(I ), i.e. every M-projective simple module is M-injective. 2. α ≤ β. 3. α-Suppl ⊆ β-Suppl. Proposition 3.4. Let M be a module. The following are equivalent: 1. S(M ) ⊆S(S ), i.e. every M-small simple module is M-singular. 2. δ ≤ γ. 3. δ-Suppl ⊆ γ-Suppl. The remaining four possible class inclusions may only be partially given in general as above. Proposition 3.5. Let M be a module. Consider the following conditions: 126 S. Crivei, D. Keskin Tütüncü / J. Algebra Comb. Discrete Appl. 4(2) (2017) 123–129 1. S(I ) ⊆S(S ), i.e. every M-injective simple module is M-singular. 2. δ ≤ α. 3. δ-Suppl ⊆ α-Suppl. Then (1) ⇔ (2) and (2) ⇒ (3). Proof. For (1) ⇒ (2) ⇒ (3) see [5, Proposition 10.27]. (2) ⇒ (1) Let S be an M-injective simple module. Then α(S) = 0. Since δ ≤ α, we have δ(S) = 0. Now there exists a nonzero homomorphism f : S −→ S ′ with S ′ simple M-singular, because δ(S) 6= S. Therefore S is M-singular. One obtains the following three propositions in a similar way. Proposition 3.6. Let M be a module. Consider the following conditions: 1. S(P) ⊆S(M ), i.e. every M-projective simple module is M-small. 2. γ ≤ β. 3. γ-Suppl ⊆ β-Suppl. Then (1) ⇔ (2) and (2) ⇒ (3). Proposition 3.7. Let M be a module. Consider the following conditions: 1. S(I ) ⊆S(P), i.e. every M-injective simple module is M-projective. 2. β ≤ α. 3. β-Suppl ⊆ α-Suppl. Then (1) ⇔ (2) and (2) ⇒ (3). Proposition 3.8. Let M be a module. Consider the following conditions: 1. S(S ) ⊆S(M ), i.e. every M-singular simple module is M-small. 2. γ ≤ δ. 3. γ-Suppl ⊆ δ-Suppl. Then (1) ⇔ (2) and (2) ⇒ (3). Obviously, the equalities S(I ) = S(P) and S(M ) = S(S ) hold if and only if the associated radicals are equal if and only if the associated proper classes coincide. In what follows we are interested in conditions under which we have equivalences in Propositions 3.5, 3.6, 3.7 and 3.8. Let τ be a preradical of σ[M]. Recall that an epimorphism f : P −→ L is called a projective τ-cover of L in σ[M] if Ker(f) ⊆ τ(P) and P is projective in σ[M] (see [1, 2.11]). Proposition 3.9. Let M be a module. Assume that the class Tγ = {X ∈ σ[M] | γ(X) = X} is closed under submodules and δγ(P) = 0 for every projective module in σ[M]. If every simple module has a projective γ-cover in σ[M], then the implication (3) ⇒ (1) holds in Proposition 3.8. 127 S. Crivei, D. Keskin Tütüncü / J. Algebra Comb. Discrete Appl. 4(2) (2017) 123–129 Proof. Assume that γ-Suppl ⊆ δ-Suppl. Let S be an M-singular simple module. So, we have δ(S) = 0. If γ(S) = 0, then S will be M-small. Suppose the contrary. Then γ(S) = S, because S is simple. Since every simple module has a projective γ-cover in σ[M], there exists an epimorphism f : P −→ S such that Ker(f) ⊆ γ(P) and P is projective in σ[M]. By [1, Lemma 2.12], f(γ(P)) = γ(S) = S. Now we have P = γ(P) + Ker(f) = γ(P). Since Tγ is closed under submodules, we have γ(Ker(f)) = Ker(f), and hence the short exact sequence 0 −→ Ker(f) −→ P −→ S −→ 0 belongs to γ-Suppl. By hypothesis, there exists a submodule L of P such that P = Ker(f) + L and L ∩ Ker(f) = δ(Ker(f)). On the other hand, δ(P) = δγ(P) = 0 and hence δ(Ker(f)) = 0. Therefore P = Ker(f)⊕L. This implies that S is M-projective, a contradiction. Therefore γ(S) = 0, and hence S is M-small. In a similar manner one obtains the following three propositions. Proposition 3.10. Let M be a module. Assume that the class Tβ = {X ∈ σ[M] | β(X) = X} is closed under submodules and αβ(P) = 0 for every projective module in σ[M]. If every simple module has a projective β-cover in σ[M], then the implication (3) ⇒ (1) holds in Proposition 3.7. Proposition 3.11. Let M be a module. Assume that the class Tγ = {X ∈ σ[M] | γ(X) = X} is closed under submodules and βγ(P) = 0 for every projective module in σ[M]. If every simple module has a projective γ-cover in σ[M], then the implication (3) ⇒ (1) holds in Proposition 3.6. Proposition 3.12. Let M be a module. Assume that the class Tδ = {X ∈ σ[M] | δ(X) = X} is closed under submodules and αδ(P) = 0 for every projective module in σ[M]. If every simple module has a projective δ-cover in σ[M], then the implication (3) ⇒ (1) holds in Proposition 3.5. Example 3.13. (see [7, Example 4.1]) Consider the category Mod-R. Let Q = ∏∞ i=1 Fi, where each Fi = Z2. Let R be the subring of Q generated by ⊕∞i=1Fi and 1Q. By [7, Example 4.1], every simple R-module has a projective δ-cover, while F0 = R/(⊕∞i=1Fi),F1,F2, . . . are the only simple R-modules and F0 is the only singular simple R-module. Therefore F1,F2, . . . are the only projective simple R-modules. This means that every simple R-module has a projective β-cover except for F0. For, assume that F0 has a projective β-cover f : P −→ F0. Since F0 is simple singular, β(F0) = F0. Therefore β(P) = P = Pβ(RR). Note that β(QR) = β ( ∞∏ i=1 Fi ) ⊆ ∞∏ i=1 β(Fi) = 0 implies that β(RR) = 0. So P = 0, which is a contradiction. Therefore F0 does not have a projective β-cover. 128 S. Crivei, D. Keskin Tütüncü / J. Algebra Comb. Discrete Appl. 4(2) (2017) 123–129 References [1] K. Al–Takhman, C. Lomp, R. Wisbauer, τ−complemented and τ−supplemented modules, Algebra Discrete Math. 3 (2006) 1–16. [2] D. Buchsbaum, A note on homology in categories, Ann. Math. 69(1) (1959) 66–74. [3] J. Clark, C. Lomp, N. Vanaja, R. Wisbauer, Lifting Modules, Frontiers in Mathematics, Birkhäuser, Basel, 2006. [4] N. V. Dung, D. V. Huynh, P. Smith, R. Wisbauer, Extending Modules, Pitman Research Notes in Mathematics, Harlow, Longman, 1994. [5] C. F. Preisser Montaño, Proper classes of short exact sequences and structure theory of modules, Ph.D. Thesis, Düsseldorf, 2010. [6] B. Stenström, Rings of Quotients, Springer, Berlin, Heidelberg, New York, 1975. [7] Y. Zhou, Generalizations of perfect, semiperfect and semiregular rings, Algebra Colloq. 7(3) (2000) 305–318. 129 http://mi.mathnet.ru/eng/adm266 http://mi.mathnet.ru/eng/adm266 http://dx.doi.org/10.2307/1970093 http://dx.doi.org/10.1007/3-7643-7573-6 http://dx.doi.org/10.1007/3-7643-7573-6 http://dx.doi.org/10.1007/978-3-642-66066-5 http://dx.doi.org/10.1007/s10011-000-0305-9 http://dx.doi.org/10.1007/s10011-000-0305-9 Introduction Preliminaries Class inclusions References