A. Çiğdem Özcan

MODULES WITH SMALL CYCLIC SUBMODULES IN THEIR INJECTIVE HULLS

ABSTRACT In this paper we study when a unital right module M over a ring R with identity has a special “small image” property we call (S*): namely, M has (S*) if every submodule N of M contains a direct summand K of M such that every cyclic submodule C of N/K is small (meaning “small in its injective hull E (C)”). If xR is small for every element x of a module M,M is said to be cosingular. In Theorem 4.4 we prove every right R-module satisfies (S*) if and only if every right R-module is the direct sum of an injective module and a cosingular module. Over a right self-injective ring R, every right R-module satisfies (S*) if and only if R is quasi-Frobenius (Theorem 5.5). It follows that over a commutative ring R, every module satisfies (S*) if and only if R is a direct product of a quasi-Frobenius ring and a cosingular ring.ABSTRACT In this paper we study when a unital right module M over a ring R with identity has a special “small image” property we call (S*): namely, M has (S*) if every submodule N of M contains a direct summand K of M such that every cyclic submodule C of N/K is small (meaning “small in its injective hull E (C)”). If xR is small for every element x of a module M,M is said to be cosingular. In Theorem 4.4 we prove every right R-module satisfies (S*) if and only if every right R-module is the direct sum of an injective module and a cosingular module. Over a right self-injective ring R, every right R-module satisfies (S*) if and only if R is quasi-Frobenius (Theorem 5.5). It follows that over a commutative ring R, every module satisfies (S*) if and only if R is a direct product of a quasi-Frobenius ring and a cosingular ring.

Semiperfect Modules with Respect to a Preradical

ABSTRACT In this article, we consider the module theoretic version of I-semiperfect rings R for an ideal I which are defined by Yousif and Zhou (2002). Let M be a left module over a ring R, N ∈ σ[M], and τ M a preradical on σ[M]. We call N τ M -semiperfect in σ[M] if for any submodule K of N, there exists a decomposition K = A ⊕ B such that A is a projective summand of N in σ[M] and B ≤ τ M (N). We investigate conditions equivalent to being a τ M -semiperfect module, focusing on certain preradicals such as Z M , Soc, and δ M . Results are applied to characterize Noetherian QF-modules (with Rad(M) ≤ Soc(M)) and semisimple modules. Among others, we prove that if every R-module M is Soc-semiperfect, then R is a Harada and a co-Harada ring.

Semiregular Modules with Respect to a Fully Invariant Submodule

Abstract Let M be a left R-module and F a submodule of M for any ring R. We call M F-semiregular if for every x ∈ M, there exists a decomposition M = A ⊕ B such that A is projective, A ≤ Rx and Rx ∩ B ≤ F. This definition extends several notions in the literature. We investigate some equivalent conditions to F-semiregular modules and consider some certain fully invariant submodules such as Z(M), Soc(M), δ(M). We prove, among others, that if M is a finitely generated projective module, then M is quasi-injective if and only if M is Z(M)-semiregular and M ⊕ M is CS. If M is projective Soc(M)-semiregular module, then M is semiregular. We also characterize QF-rings R with J(R)2 = 0.

The Torsion Theory Cogenerated by δ–M-Small Modules and GCO-Modules

Let M be a module and K a submodule of a module N in σ[M]. We call K a δ–M-small submodule of N if whenever N = K + L, N/L is M-singular for any submodule L of N, we have N = L. Also we call N a δ–M-small module if N is a δ–M-small submodule of its M-injective hull. In this article, we consider , the reject of 𝒟 ℳ in N, where 𝒟 ℳ is the class of all δ–M-small modules. We investigate the properties of and consider the torsion theory τδ V in σ[M] cogenerated by 𝒟 ℳ. We compare the τδ V and the torsion theory τ V cogenerated by M-small modules and finally we give a characterization of GCO-modules.

A Generalization of Semiregular and Semiperfect Modules

Let U be a submodule of a module M. We call U a strongly lifting submodule of M if whenever M/U=(A+U)/U ⊕ (B+U)/U, then M=P ⊕ Q such that P ≤ A, (A+U)/U=(P+U)/U and (B+U)/U=(Q+U)/U. This definition is a generalization of strongly lifting ideals defined by Nicholson and Zhou. In this paper, we investigate some properties of strongly lifting submodules and characterize U-semiregular and U-semiperfect modules by using strongly lifting submodules. Results are applied to characterize rings R satisfying that every (projective) left R-module M is τ (M)-semiperfect for some preradicals τ such as Rad, Z2 and δ.

ON JACOBSON AND NIL RADICALS RELATED TO POLYNOMIAL RINGS

Abstract. This note is concerned with examining nilradicals and Jacob-son radicals of polynomial rings when related factor rings are Armendariz.Especially we elaborate upon a well-known structural property of Armen-dariz rings, bringing into focus the Armendariz property of factor rings byJacobson radicals. We show that J(R[x]) = J(R)[x] if and only if J(R) isnil when a given ring R is Armendariz, where J(A) means the Jacobsonradical of a ring A. A ring will be called feckly Armendariz if the factorring by the Jacobson radical is an Armendariz ring. It is shown that thepolynomial ring over an Armendariz ring is feckly Armendariz, in spiteof Armendariz rings being not feckly Armendariz in general. It is alsoshown that the feckly Armendariz property does not go up to polynomialrings. 1. On radicals whenfactor rings are ArmendarizThroughout this note every ring is associative with identity unless other-wise stated. For a ring R, J(R), N ∗ (R), N ∗ (R), N 0 (R) and N(R) denotethe Jacobson radical, the prime radical, the upper nilradical (i.e., sum of allnil ideals), the Wedderburn radical (i.e., the sum of all nilpotent ideals), andthe set of all nilpotent elements in R, respectively. Following [1, p. 130], asubset of R is said to be locally nilpotent if its finitely generated subringsare nilpotent. Also due to [1, p. 130], the Levitzki radical of R, written bysσ(R), means the sum of all locally nilpotent ideals of R. It is well-knownthat N

On internally cancellable rings

The concept of internally cancellable rings has been extensively studied in the literature. This paper seeks to continue the study of these rings and find some new characterizations. It is proved that R is “IC”, if and only if for each regular element a ∈ R, and idempotent element b ∈ R with Ra + Rb = R, there exists x ∈ R such that a + xb is a unit (alternatively, unit-regular element) in R and aR ∩ xR = 0. In case the ring R has the summand sum property, we indicate that R is IC, if and only if for each regular element a ∈ R, and element b ∈ R with Ra + Rb = R, there exists an idempotent e ∈ R, such that a + eb is a unit in R and aR ∩ eR = 0.

A Generalization of Semiregular and Almost Principally Injective Rings

In this article, we call a ring R right almost I-semiregular for an ideal I of R if for any a ∈ R, there exists a left R-module decomposition lRrR(a) = P ⊕ Q such that P ⊆ Ra and Q ∩ Ra ⊆ I, where l and r are the left and right annihilators, respectively. This generalizes the right almost principally injective rings defined by Page and Zhou, I-semiregular rings defined by Nicholson and Yousif, and right generalized semiregular rings defined by Xiao and Tong. We prove that R is I-semiregular if and only if for any a ∈ R, there exists a decomposition lRrR(a) = P ⊕ Q, where P = Re ⊆ Ra for some e2 = e ∈ R and Q ∩ Ra ⊆ I. Among the results for right almost I-semiregular rings, we show that if I is the left socle Soc(RR) or the right singular ideal Z(RR) or the ideal Z(RR) ∩ δ(RR), where δ(RR) is the intersection of essential maximal left ideals of R, then R being right almost I-semiregular implies that R is right almost J-semiregular for the Jacobson radical J of R. We show that δl(eRe) = e δ(RR)e for any idempotent e of R satisfying ReR = R and, for such an idempotent, R being right almost δ(RR)-semiregular implies that eRe is right almost δl(eRe)-semiregular.

δ-M-Small and δ-Harada Modules

Let M be a right R-module and N ∈ σ[M]. A submodule K of N is called δ-M-small if, whenever N = K + X with N/X M-singular, we have N = X. N is called a δ-M-small module if N≅ K, K is δ-M-small in L for some K, L ∈ σ[M]. In this article, we prove that if M is a finitely generated self-projective generator in σ[M], then M is a Noetherian QF-module if and only if every module in σ[M] is a direct sum of a projective module in σ[M] and a δ-M-small module. As a generalization of a Harada module, a module M is called a δ-Harada module if every injective module in σ[M] is δ M -lifting. Some properties of δ-Harada modules are investigated and a characterization of a Harada module is also obtained.

C4- and D4-Modules via perspective direct summands

ABSTRACT In this article, we study the C4- and D4-modules in terms of perspective direct summands, providing new characterizations and results. Endomorphism rings of C4-modules and extensions of right C4-rings are also investigated. Decompositions of C4-modules with restricted ACC on direct summands and D4-modules with restricted DCC on direct summands are obtained.