scholarly journals s.Baer and s.Rickart Modules

2015 ◽  
Vol 14 (08) ◽  
pp. 1550131 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Richard L. LeBlanc
Keyword(s):  
Nonempty Subset ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Torsion Theory ◽  
The Other ◽  
Projective Modules ◽  
Closure Properties ◽  
Triangular Matrix Ring ◽  
The Right ◽  
Torsion Modules

In this paper, we study module theoretic definitions of the Baer and related ring concepts. We say a module is s.Baer if the right annihilator of a nonempty subset of the module is generated by an idempotent in the ring. We show that s.Baer modules satisfy a number of closure properties. Under certain conditions, a torsion theory is established for the s.Baer modules, and we provide examples of s.Baer torsion modules and modules with a nonzero s.Baer radical. The other principal interest of this paper is to provide explicit connections between s.Baer modules and projective modules. Among other results, we show that every s.Baer module is an essential extension of a projective module. Additionally, we prove, with limited and natural assumptions, that in a generalized triangular matrix ring every s.Baer submodule of the ring is projective. As an application, we show that every prime ring with a minimal right ideal has the strong summand intersection property. Numerous examples are provided to illustrate, motivate, and delimit the theory.

Projection invariant extending rings

2016 ◽  
Vol 15 (07) ◽  
pp. 1650121 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Adnan Tercan ◽  
Canan C. Yucel
Keyword(s):  
Direct Summand ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Trivial Extension ◽  
Matrix Rings ◽  
Finite Matrix ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
The Right ◽  
Upper Triangular Matrix Ring

A ring [Formula: see text] is said to be right [Formula: see text]-extending if every projection invariant right ideal of [Formula: see text] is essential in a direct summand of [Formula: see text]. In this article, we investigate the transfer of the [Formula: see text]-extending condition between a ring [Formula: see text] and its various ring extensions. More specifically, we characterize the right [Formula: see text]-extending generalized triangular matrix rings; and we show that if [Formula: see text] is [Formula: see text]-extending, then so is [Formula: see text] where [Formula: see text] is an overring of [Formula: see text] which is an essential extension of [Formula: see text], an [Formula: see text] upper triangular matrix ring of [Formula: see text], a column finite or column and row finite matrix ring over [Formula: see text], or a certain type of trivial extension of [Formula: see text].

scholarly journals Relative Gorenstein Dimensions over Triangular Matrix Rings

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2676
Author(s):  
Driss Bennis ◽  
Rachid El Maaouy ◽  
Juan Ramón García Rozas ◽  
Luis Oyonarte
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Projective Dimension ◽  
Global Dimension ◽  
Projective Modules ◽  
Matrix Rings ◽  
Weakly Compatible ◽  
Gorenstein Algebra ◽  
Finite Projective Dimension ◽  
Triangular Matrix Ring

Let A and B be rings, U a (B,A)-bimodule, and T=A0UB the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We first study how to construct w-tilting (tilting, semidualizing) over T using the corresponding ones over A and B. We show that when U is relative (weakly) compatible, we are able to describe the structure of GC-projective modules over T. As an application, we study when a morphism in T-Mod is a special GCP(T)-precover and when the class GCP(T) is a special precovering class. In addition, we study the relative global dimension of T. In some cases, we show that it can be computed from the relative global dimensions of A and B. We end the paper with a counterexample to a result that characterizes when a T-module has a finite projective dimension.

Weakly principally quasi-Baer rings

2015 ◽  
Vol 15 (01) ◽  
pp. 1650002 ◽  
Author(s):  
A. Majidinya ◽  
A. Moussavi
Keyword(s):  
Prime Ideal ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Direct Products ◽  
Product Representation ◽  
Triangular Matrix Ring ◽  
Baer Rings ◽  
Upper Triangular Matrix ◽  
The Right ◽  
Minimal Prime

We say a ring R with unity is weakly principally quasi-Baer or simply (weakly p.q.-Baer) if the right annihilator of a principal right ideal is right s-unital by right semicentral idempotents, which implies that R modulo the right annihilator of any principal right ideal is flat. It is proven that the weakly p.q.-Baer property is inherited by polynomial extensions and includes both left p.q.-Baer rings and right p.q.-Baer rings and is closed under direct products and Morita invariance. A ring R is weakly p.q.-Baer if and only if the upper triangular matrix ring Tn(R) is weakly p.q.-Baer. We extend a theorem of Kist for commutative Baer rings to weakly p.q.-Baer rings for which every prime ideal contains a unique minimal prime ideal without using topological arguments. It is also shown that there is an important subclass of weakly p.q.-Baer rings which have a nontrivial subdirect product representation.

scholarly journals ON WEAKLY RIGID RINGS

2009 ◽  
Vol 51 (3) ◽  
pp. 425-440 ◽  
Author(s):  
A. R. NASR-ISFAHANI ◽  
A. MOUSSAVI
Keyword(s):  
Prime Ring ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
The Other ◽  
Rigid Ring ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
Upper Triangular Matrix Ring

AbstractLet R be a ring with a monomorphism α and an α-derivation δ. We introduce (α, δ)-weakly rigid rings which are a generalisation of α-rigid rings and investigate their properties. Every prime ring R is (α, δ)-weakly rigid for any automorphism α and α-derivation δ. It is proved that for any n, a ring R is (α, δ)-weakly rigid if and only if the n-by-n upper triangular matrix ring Tn(R) is (, )-weakly rigid if and only if Mn(R) is (, )-weakly rigid. Moreover, various classes of (α, δ)-weakly rigid rings is constructed, and several known results are extended. We show that for an (α, δ)-weakly rigid ring R, and the extensions R[x], R[[x]], R[x; α, δ], R[x, x−1; α], R[[x; α]], R[[x, x−1; α]], the ring R is quasi-Baer if and only if the extension over R is quasi-Baer. It is also proved that for an (α, δ)-weakly rigid ring R, if any one of the rings R, R[x], R[x; α, δ] and R[x, x−1; α] is left principally quasi-Baer, then so are the other three. Examples to illustrate and delimit the theory are provided.

The set of singular elements of a ring with respect to a module

2018 ◽  
Vol 13 (03) ◽  
pp. 2050050
Author(s):  
A. Farzi–Safarabadi ◽  
R. Beyranvand
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Semisimple Module ◽  
Prime Module ◽  
Triangular Matrix Ring ◽  
Singular Ideal ◽  
The Right

Let [Formula: see text] be a ring and [Formula: see text] be a right [Formula: see text]-module. In this paper, we introduce the set [Formula: see text] for some essential submodule [Formula: see text] of [Formula: see text] of singular elements of[Formula: see text] with respect to[Formula: see text] , and we investigate the properties of it. For example, it is shown that [Formula: see text] is an ideal of [Formula: see text] and [Formula: see text]. Also if [Formula: see text] is a semiprime right Goldie ring, then [Formula: see text], where [Formula: see text] is the right singular ideal of [Formula: see text]. We prove that if [Formula: see text] is a semisimple module or a prime module, then [Formula: see text]. For any submodule [Formula: see text] of [Formula: see text], we have [Formula: see text] and if [Formula: see text], then [Formula: see text]. We show that [Formula: see text] and [Formula: see text]. In the end, the singular elements of some rings with respect to the formal triangular matrix ring are investigated.

PURE PROJECTIVITY AND PURE INJECTIVITY OVER FORMAL TRIANGULAR MATRIX RINGS

2012 ◽  
Vol 11 (06) ◽  
pp. 1250107 ◽  
Author(s):  
A. HAGHANY ◽  
M. MAZROOEI ◽  
M. R. VEDADI
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Matrix Rings ◽  
Triangular Matrix Ring

Over a formal triangular matrix ring we study pure injective, pure projective and locally coherent modules. Some applications are then given, in particular the (J-)coherence of the ring [Formula: see text] is characterized whenever BM is flat.

Integer-valued polynomials over block matrix algebras

2019 ◽  
Vol 19 (03) ◽  
pp. 2050053
Author(s):  
J. Sedighi Hafshejani ◽  
A. R. Naghipour ◽  
M. R. Rismanchian
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Block Matrix ◽  
Quotient Field ◽  
Matrix Rings ◽  
Matrix Algebras ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
Full Matrix Ring ◽  
Upper Triangular Matrix Ring

In this paper, we state a generalization of the ring of integer-valued polynomials over upper triangular matrix rings. The set of integer-valued polynomials over some block matrix rings is studied. In fact, we consider the set of integer-valued polynomials [Formula: see text] for each [Formula: see text], where [Formula: see text] is an integral domain with quotient field [Formula: see text] and [Formula: see text] is a block matrix ring between upper triangular matrix ring [Formula: see text] and full matrix ring [Formula: see text]. In fact, we have [Formula: see text]. It is known that the sets of integer-valued polynomials [Formula: see text] and [Formula: see text] are rings. We state some relations between the rings [Formula: see text] and the partitions of [Formula: see text]. Then, we show that the set [Formula: see text] is a ring for each [Formula: see text]. Further, it is proved that if the ring [Formula: see text] is not Noetherian then the ring [Formula: see text] is not Noetherian, too. Finally, some properties and relations are stated between the rings [Formula: see text], [Formula: see text] and [Formula: see text].

scholarly journals Abel Rings and Super-Strongly Clean Rings

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Yinchun Qu ◽  
Junchao Wei
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Clean Rings ◽  
Clean Ring ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
Upper Triangular Matrix Ring

Abstract In this note, we first show that a ring R is Abel if and only if the 2 × 2 upper triangular matrix ring over R is quasi-normal. Next, we give the notion of super-strongly clean ring (that is, an Abel clean ring), which is inbetween uniquely clean rings and strongly clean rings. Some characterizations of super-strongly clean rings are given.

Hulls of Ring Extensions

2010 ◽  
Vol 53 (4) ◽  
pp. 587-601 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Jae Keol Park ◽  
S. Tariq Rizvi
Keyword(s):  
Group Ring ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Affirmative Answer ◽  
Infinite Matrix ◽  
Matrix Rings ◽  
Morita Equivalent ◽  
Semiprime Rings ◽  
Ring Extensions ◽  
The Right

AbstractWe investigate the behavior of the quasi-Baer and the right FI-extending right ring hulls under various ring extensions including group ring extensions, full and triangular matrix ring extensions, and infinite matrix ring extensions. As a consequence, we show that for semiprime rings R and S, if R and S are Morita equivalent, then so are the quasi-Baer right ring hulls of R and S, respectively. As an application, we prove that if unital C*-algebras A and B are Morita equivalent as rings, then the bounded central closure of A and that of B are strongly Morita equivalent as C*-algebras. Our results show that the quasi-Baer property is always preserved by infinite matrix rings, unlike the Baer property. Moreover, we give an affirmative answer to an open question of Goel and Jain for the commutative group ring A[G] of a torsion-free Abelian group G over a commutative semiprime quasi-continuous ring A. Examples that illustrate and delimit the results of this paper are provided.

Homological dimensions of special modules over formal triangular matrix rings

2021 ◽  
pp. 2250146
Author(s):  
Lixin Mao
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Matrix Rings ◽  
Special Formula ◽  
Homological Dimensions ◽  
Triangular Matrix Ring

Let [Formula: see text] be a formal triangular matrix ring, where [Formula: see text] and [Formula: see text] are rings and [Formula: see text] is a [Formula: see text]-bimodule. We give some computing formulas of homological dimensions of special [Formula: see text]-modules. As an application, we describe the structures of [Formula: see text]-tilting left [Formula: see text]-modules.

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