quasi-Baer Modules Research Articles

On the P.Q.-Baer Skew Generalized Power Series Modules

For a ring [Formula: see text] and a strictly totally ordered monoid [Formula: see text], let [Formula: see text] be a monoid homomorphism and [Formula: see text] an [Formula: see text]-weakly rigid right [Formula: see text]-module (i.e., for any elements [Formula: see text], [Formula: see text] and [Formula: see text], [Formula: see text] if and only if [Formula: see text]), where [Formula: see text] is the ring of ring endomorphisms of [Formula: see text]. It is shown that the skew generalized power series module [Formula: see text] is a principally quasi-Baer module if and only if the annihilator of every submodule generated by an [Formula: see text]-indexed subset of [Formula: see text] is generated by an idempotent as a right ideal of [Formula: see text]. As a consequence we deduce that for an [Formula: see text]-weakly rigid ring [Formula: see text], the skew generalized power series ring [Formula: see text] is right principally quasi-Baer if and only if [Formula: see text] is right principally quasi-Baer and any [Formula: see text]-indexed subset of right semicentral idempotents in [Formula: see text] has a generalized [Formula: see text]-indexed join in [Formula: see text]. The range of previous results in this area is expanded by these results.

The structure of piecewise prime modules

The purpose of this paper is to further the study of quasi-Baer modules by investigating the structure of a special class of quasi-Baer modules. As a module theoretic analog of a piecewise prime ring, we define and characterize a piecewise prime module (simply, a PWP module) via its endomorphism ring. Although it is well known that eRe is a PWP ring for a left (right) semicentral idempotent or a full idempotent e in a PWP ring R, it is still an open question whether this holds true when e is an arbitrary idempotent. We give an affirmative answer to this question. As a consequence, we prove that every direct summand of a PWP module is a PWP module. It is shown that any direct sum of copies of a PWP module is always a PWP module. Consequently, every column (and row) finite matrix ring over a PWP ring is a PWP ring. We obtain a complete structure theorem for PWP modules and show that endoprime submodules are the building blocks of the PWP modules. Applications and examples illustrating our results are provided.

Some Results on C-retractable Modules

An R-module M is called c-retractable if there exists a nonzero homomorphism from M to any of its nonzero complement submodules. In this paper, we provide some new results of c- retractable modules. It is shown that every projective module over a right SI-ring is c-retractable. A dual Baer c-retractable module is a direct sum of a Z2-torsion module and a module which is a direct sum of nonsingular uniform quasi-Baer modules whose endomorphism rings are semi- local quasi-Baer. Conditions are found under which, a c-retractable module is extending, quasi-continuous, quasi-injective and retractable. Also, it is shown that a locally noetherian c-retractable module is homo-related to a direct sum of uniform modules. Finally, rings over which every c-retractable is a C4-module are determined.

Some Results on C-retractable Modules

An R-module M is called c-retractable if there exists a nonzero homomorphism from M to any of its nonzero complement submodules. In this paper, we provide some new results of c- retractable modules. It is shown that every projective module over a right SI-ring is c-retractable. A dual Baer c-retractable module is a direct sum of a Z2-torsion module and a module which is a direct sum of nonsingular uniform quasi-Baer modules whose endomorphism rings are semi- local quasi-Baer. Conditions are found under which, a c-retractable module is extending, quasi-continuous, quasi-injective and retractable. Also, it is shown that a locally noetherian c-retractable module is homo-related to a direct sum of uniform modules. Finally, rings over which every c-retractable is a C4-module are determined.

π-endo Baer modules

Let N be a submodule of a right R-module MR, and Then N is said to be projection invariant in M, denoted by if for all We call MR π-endo Baer, denoted π-e.Baer, if for each there exists such that where denotes the left annihilator of N in H. We show that this class of modules lies strictly between the classes of Baer and quasi-Baer modules introduced in 2004 by Rizvi and Roman. Several structural properties are developed. In contrast to the Baer modules of Rizvi and Roman, the free modules of a Baer ring are π-e.Baer. Moreover, (co-) nonsingularity conditions are introduced which enable us to extend the Chatters-Khuri result (connecting the extending and Baer conditions in a ring) to modules. We provide examples to illustrate and delimit our results.

Principally quasi-Baer modules and their generalizations

The purpose of this article is to further the study of principally quasi-Baer modules and -principally quasi-Baer modules as these properties play an important role in the study of quasi-Baer module. First, we provide some basic results and a characterization of a principally quasi-Baer (simply, p.q.-Baer) module in terms of its endomorphism ring by using the pq-local-retractable property. In addition, we fully characterize when a finite direct sum of arbitrary p.q.-Baer modules is p.q.-Baer. Next, we obtain characterizations and properties of -principally quasi-Baer (simply, -p.q.-Baer) modules. Examples which show that the notion of an -p.q.-Baer module is distinct from that of a p.q.-Baer module are provided. It is shown that every direct summand of an -p.q.-Baer module inherits the property. Furthermore, we obtain that every direct sum of copies of an -p.q.-Baer module is an -p.q.-Baer module. We provide conditions when (-)p.q.-Baer modules become quasi-Baer modules. In particular, if every direct sum of copies of a module M is p.q.-Baer then the module M is a quasi-Baer module.

Endoprime modules and their direct sums

The purpose of this paper is to further study the endoprime modules as one of the special classes of quasi-Baer modules. As a module theoretic analogue of a prime ring, we characterize an endoprime module via its endomorphism ring and a weak retractability condition. It is shown that any direct summand of an endoprime module is an endoprime module. A characterization is obtained when a direct sum of endoprime modules is an endoprime module. It is well known that every prime ring is semicentral reduced. We prove that a column (and row) finite matrix ring over a semicentral reduced ring is also a semicentral reduced ring. Consequently, it is shown that a column (and row) finite matrix ring over a prime ring is prime. Applications and examples illustrating our results are provided.

A Counterexample for a Problem on Quasi Baer Modules

In this note we answer two questions on quasi-Baer modules raised by Lee and Rizvi in J.Algebra (2016).

Quasi-Baer module hulls and applications

Let V be a module with S=End(V). V is called a quasi-Baer module if for each ideal J of S, rV(J)=eV for some e2=e∈S. On the other hand, V is called a Rickart module if for each ϕ∈S, Ker(ϕ)=eV for some e2=e∈S. For a module N, the quasi-Baer module hull qB(N) (resp., the Rickart module hull Ric(N)) of N, if it exists, is the smallest quasi-Baer (resp., Rickart) overmodule, in a fixed injective hull E(N) of N. In this paper, we initiate the study of quasi-Baer and Rickart module hulls. When a ring R is semiprime and ideal intrinsic over its center, it is shown that every finitely generated projective R-module has a quasi-Baer hull. Let R be a Dedekind domain with F its field of fractions and let {Ki|i∈Λ} be any set of R-submodules of FR. For an R-module MR with AnnR(M)≠0, we show that MR⊕(⨁i∈ΛKi)R has a quasi-Baer module hull if and only if MR is semisimple. This quasi-Baer hull is explicitly described. An example such that MR⊕(⨁i∈ΛKi)R has no Rickart module hull is constructed. If N is a module over a Dedekind domain for which N/t(N) is projective and AnnR(t(N))≠0, where t(N) is the torsion submodule of N, we show that the quasi-Baer hull qB(N) of N exists if and only if t(N) is semisimple. We prove that the Rickart module hull also exists for such modules N. Furthermore, we provide explicit constructions of qB(N) and Ric(N) and show that in this situation these two hulls coincide. Among applications, it is shown that if N is a finitely generated module over a Dedekind domain, then N is quasi-Baer if and only if N is Rickart if and only if N is Baer if and only if N is semisimple or torsion-free. For a direct sum NR of finitely generated modules, where R is a Dedekind domain, we show that N is quasi-Baer if and only if N is Rickart if and only if N is semisimple or torsion-free. Examples exhibiting differences between the notions of a Baer hull, a quasi-Baer hull, and a Rickart hull of a module are presented. Various explicit examples illustrating our results are constructed.

Direct sums of quasi-Baer modules

While the research work on quasi-Baer rings has been quite extensive in existing literature, the study of quasi-Baer modules introduced in 2004 remains quite limited with only little work available on the notion. In this paper, we extend the study of quasi-Baer modules. A characterization of a quasi-Baer module in terms of its endomorphism ring is obtained. We provide a complete characterization of arbitrary direct sums of quasi-Baer modules to be quasi-Baer. We also show characterizations of some important properties of quasi-Baer modules. Examples which delineate the concepts and the results are provided.

Baer and Quasi-Baer Modules over Some Classes of Rings

We study Baer and quasi-Baer modules over some classes of rings. We also introduce a new class of modules called AI-modules, in which the kernel of every nonzero endomorphism is contained in a proper direct summand. The main results obtained here are: (1) A module is Baer iff it is an AI-module and has SSIP. (2) For a perfect ring R, the direct sum of Baer modules is Baer iff R is primary decomposable. (3) Every injective R-module is quasi-Baer iff R is a QI-ring.

MODULES WITH FI-EXTENDING HULLS

AbstractIt is shown that every finitely generated projective module PR over a semiprime ring R has the smallest FI-extending essential module extension $H_{\frak{FI}}(P_R)$ (called the absolute FI-extending hull of PR) in a fixed injective hull of PR. This module hull is explicitly described. It is proved that $\widehat{Q}_{\frak{FI}}({\rm End}(P_R))\cong {\rm End}(H_{\frak{FI}}(P_R))$, where $\widehat{Q}_{\frak{FI}}({\rm End}(P_R))$ is the smallest right FI-extending right ring of quotients of End(PR) (in a fixed maximal right ring of quotients of End(PR). Moreover, we show that a finitely generated projective module PR over a semiprime ring R is FI-extending if and only if it is a quasi-Baer module and if and only if End(PR) is a quasi-Baer ring. An application of this result to C*-algebras is considered. Various examples which illustrate and delimit the results of this paper are provided.

ON QUASI-ARMENDARIZ MODULES

n this paper, we introduce the concept of a ($\alpha$-) quasi-Armendariz module, principally quasi-Baer module and syudy its some properties. In particular, we show: (1) For an $\alpha$-quasi-Armendariz module $M_R$, $M_R$ is a principally quasi-Baer module if and only if $M[x;\alpha]_{R[x;\alpha]}$ is a principally quasi-Baer module. (2) A necessary and sufficient condition for a trivial extensions to be quasi-Armendariz is obtained. Consequently, new families of quasi-Armendariz rings are presented.

SPECIAL PROPERTIES OF MODULES OF GENERALIZED POWER SERIES

Let $R$ be a ring, $M$ a right $R$-module and $(S,\leq)$ a strictly ordered monoid. In this paper, a necessary and sufficient condition is given for modules under which $[[M^{S,\leq}]]_{[[R^{S,\leq}]]}$, the module of generalized power series with coefficients in $M$ and exponents in $S$ is a reduced, Baer, PP. quasi-Baer module, respectively.

ON QUASI-ARMENDARIZ MODULES

n this paper, we introduce the concept of a ($\alpha$-) quasi-Armendariz module, principally quasi-Baer module and syudy its some properties. In particular, we show: (1) For an $\alpha$-quasi-Armendariz module $M_R$, $M_R$ is a principally quasi-Baer module if and only if $M[x;\alpha]_{R[x;\alpha]}$ is a principally quasi-Baer module. (2) A necessary and sufficient condition for a trivial extensions to be quasi-Armendariz is obtained. Consequently, new families of quasi-Armendariz rings are presented.

SPECIAL PROPERTIES OF MODULES OF GENERALIZED POWER SERIES

Let $R$ be a ring, $M$ a right $R$-module and $(S,\leq)$ a strictly ordered monoid. In this paper, a necessary and sufficient condition is given for modules under which $[[M^{S,\leq}]]_{[[R^{S,\leq}]]}$, the module of generalized power series with coefficients in $M$ and exponents in $S$ is a reduced, Baer, PP. quasi-Baer module, respectively.

Baer and Quasi-Baer Modules#

We introduce the notions of Baer and quasi-Baer properties in a general module theoretic setting. A module M is called (quasi-) Baer if the right annihilator of a (two-sided) left ideal of End(M) is a direct summand of M. We show that a direct summand of a (quasi-) Baer module inherits the property and every finitely generated abelian group is Baer exactly if it is semisimple or torsion-free. Close connections to the (FI-) extending property are investigated and it is shown that a module M is (quasi-) Baer and (FI-) 𝒦-cononsingular if and only if it is (FI-) extending and (FI-) 𝒦-nonsingular. We prove that an arbitrary direct sum of mutually subisomorphic quasi-Baer modules is quasi-Baer and every free (projective) module over a quasi-Baer ring is a quasi-Baer module. Among other results, we also show that the endomorphism ring of a (quasi-) Baer module is a (quasi-) Baer ring, while the converse is not true in general. Applications of results are provided.