TRIANGULAR MATRIX REPRESENTATIONS OF SEMIPRIMARY RINGS

2002 ◽  
Vol 01 (02) ◽  
pp. 123-131 ◽  
Author(s):  
GARY F. BIRKENMEIER ◽  
JIN YONG KIM ◽  
JAE KEOL PARK
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Prime Ideals ◽  
Matrix Representations ◽  
Artinian Rings ◽  
Baer Ring ◽  
Proper Subclass ◽  
Triangular Matrix Ring ◽  
Baer Rings

In this paper we characterize internally a TSA ring (i.e. a generalized triangular matrix ring with simple Artinian rings on the diagonal) in terms of its prime ideals. Also we show that the class of semiprimary quasi-Baer rings is a proper subclass of the class of TSA rings. Moreover, we generalize results of Harada, Small, and Teply on semiprimary rings. Finally we prove that under certain cardinality conditions a semiprimary quasi-Baer ring becomes semisimple Artinian.

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.

On reduced rank of triangular matrix rings

2015 ◽  
Vol 14 (04) ◽  
pp. 1550059
Author(s):  
Abigail C. Bailey ◽  
John A. Beachy
Keyword(s):  
Homomorphic Image ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Matrix Rings ◽  
Triangular Matrices ◽  
Reduced Rank ◽  
Artinian Rings ◽  
Triangular Matrix Ring ◽  
Lower Triangular ◽  
Lower Triangular Matrices

We determine conditions under which a generalized triangular matrix ring has finite reduced rank, in the general torsion-theoretic sense. These are applied to characterize certain orders in Artinian rings, and to show that if each homomorphic image of a ring S has finite reduced rank, then so does the ring of lower triangular matrices over S.

scholarly journals A note on commutative Baer rings

1972 ◽  
Vol 14 (3) ◽  
pp. 257-263 ◽  
Author(s):  
T. P. Speed
Keyword(s):  
Semiprime Ring ◽  
Algebraic Approach ◽  
Direct Decomposition ◽  
Prime Ideals ◽  
Baer Ring ◽  
Baer Rings ◽  
New Construction ◽  
Minimal Prime

In this note we study commutative Baer rings, uniting the abstract algebraic approach with the approach of [3] using minimal prime ideals. Some new characterisations of this class of rings are obtained, relations between the minimal prime ideals of a commutative Baer ring B and its algebra EB of idempotents are considered, and some results concerning the direct decomposition of commutative Baer rings are given. We then study Baer ideals, and finally state without proof a new construction of the Baer extension of a commutative semiprime ring.

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.

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.

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 Strongly J-Clean Skew Triangular Matrix Rings

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Yosum Kurtulmaz
Keyword(s):  
Local Ring ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Matrix Rings ◽  
Arbitrary Ring ◽  
Triangular Matrix Ring

Abstract Let R be an arbitrary ring with identity. An element a ∈ R is strongly J-clean if there exist an idempotent e ∈ R and element w ∈ J(R) such that a = e + w and ew = ew. A ring R is strongly J-clean in case every element in R is strongly J-clean. In this note, we investigate the strong J-cleanness of the skew triangular matrix ring Tn(R, σ) over a local ring R, where σ is an endomorphism of R and n = 2, 3, 4.

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