Sait Halicioglu

Modules in Which Inverse Images of Some Submodules are Direct Summands

Let R be an arbitrary ring with identity and M a right R-module with S = EndR(M). Let F be a fully invariant submodule of M. We call M an F-inverse split module if f−1(F) is a direct summand of M for every f ∈ S. This work is devoted to investigation of various properties and characterizations of an F-inverse split module M and to show, among others, the following results: (1) the module M is F-inverse split if and only if M = F ⊕ K where K is a Rickart module; (2) for every free R-module M, there exists a fully invariant submodule F of M such that M is F-inverse split if and only if for every projective R-module M, there exists a fully invariant submodule F of M such that M is F-inverse split; and (3) Every R-module M is Z2(M)-inverse split and Z2(M) is projective if and only if R is semisimple.

Rickart modules relative to singular submodule and dual Goldie torsion theory

Let R be an arbitrary ring with identity and M a right R-module with the ring S = EndR(M) of endomorphisms of M. The notion of an F-inverse split module M, where F is a fully invariant submodule of M, is defined and studied by the present authors. This concept produces Rickart submodules of modules in the sense of Lee, Rizvi and Roman. In this paper, we consider the submodule F of M as Z(M) and Z∗(M), and investigate some properties of Z(M)-inverse split modules and Z∗(M)-inverse split modules M. Results are applied to characterize rings R for which every free (projective) right R-module M is F-inverse split for the preradicals such as Z(⋅) and Z∗(⋅).

Feckly reduced rings

Let R be a ring with identity and J(R) denote the Jacobson radical of R. In this paper, we introduce a new class of rings called feckly reduced rings. The ring R is called feckly reduced if R=J(R) is a reduced ring. We investigate relations between feckly reduced rings and other classes of rings. We obtain some characterizations of being a feckly reduced ring. It is proved that a ring R is feckly reduced if and only if every cyclic projective R-module has a feckly reduced endomorphism ring. Among others we show that every left Artinian ring is feckly reduced if and only if it is 2-primal, R is feckly reduced if and only if T(R;R) is feckly reduced if and only if R[x]= is feckly reduced.

Strongly Large Module Extensions

Abstract In this paper, we introduce strongly large submodules and investigate their properties. A submodule N of a right R-module M is said to be strongly large in case for any m ∈ M, s ∈ R with ms ̸= 0 there exists an r ∈ R such that mr ∈ N and mrs ̸= 0. In this note, we also define and study strongly large closed submodules and strongly large complement submodules

A dual approach to the theory of inverse split modules

Let R be an arbitrary ring with identity, M a right R-module and F a fully invariant submodule of M. The notion of an F-inverse split module M has been defined and studied by the present authors recently. In this paper, we introduce its dual notion, namely, dualF-inverse split moduleM . This work is devoted to investigation of various properties and characterizations of a dual F-inverse split module M. We include applications for rings and cosingular submodules. We also deal with the notion of relatively dual inverse splitness to investigate direct sums of dual inverse split modules.

Modules having Baer summands

ABSTRACT Let R be an arbitrary ring with identity and M a right R-module with S = EndR(M). Let F be a fully invariant submodule of M and I−1(F) denotes the set {m∈M:Im⊆F} for any subset I of S. The module M is called F-Baer if I−1(F) is a direct summand of M for every left ideal I of S. This work is devoted to the investigation of properties of F-Baer modules. We use F-Baer modules to decompose a module into two parts consists of a Baer module and a module determined by fully invariant submodule F, namely, for a module M, we show that M is F-Baer if and only if M = F⊕N where N is a Baer module. By using F-Baer modules, we obtain some new results for Baer rings.

Quasipolar Subrings of 3 x 3 Matrix Rings

Abstract An element a of a ring R is called quasipolar provided that there exists an idempotent p ∈ R such that p ∈ comm2(a), a + p ∈ U (R) and ap ∈ Bqnil. A ring R is quasipolar in case every element in R is quasipolar. In this paper, we determine conditions under which subrings of 3 x 3 matrix rings over local rings are quasipolar. Namely, if R. is a bleached local ring, then we prove that T3 (R) is quasipolar if and only if R is uniquely bleached. Furthermore, it is shown that Tn(R) is quasipolar if and only if Tn(R[[x]]) is quasipolar for any positive integer