scholarly journals Classical quotient rings of generalized matrix rings

1995 ◽  
Vol 18 (2) ◽  
pp. 311-316 ◽  
Author(s):  
David G. Poole ◽  
Patrick N. Stewart
Keyword(s):  
Associative Ring ◽  
Matrix Ring ◽  
Quotient Ring ◽  
Matrix Rings ◽  
Finite Set ◽  
Generalized Matrix ◽  
Quotient Rings

An associative ringRwith identity is a generalized matrix ring with idempotent setEifEis a finite set of orthogonal idempotents ofRwhose sum is1. We show that, in the presence of certain annihilator conditions, such a ring is semiprime right Goldie if and only ifeReis semiprime right Goldie for alle∈E, and we calculate the classical right quotient ring ofR.

A Generalization of the Concept of a Ring of Quotients

1971 ◽  
Vol 14 (4) ◽  
pp. 517-529 ◽  
Author(s):  
John K. Luedeman
Keyword(s):  
Left Ideal ◽  
Quotient Ring ◽  
Original Method ◽  
Duke Math ◽  
Injective Hull ◽  
Direct Products ◽  
Left Module ◽  
Matrix Rings ◽  
Ring Of Quotients ◽  
Quotient Rings

AbstractSanderson (Canad. Math. Bull., 8 (1965), 505–513), considering a nonempty collection Σ of left ideals of a ring R, with unity, defined the concepts of “Σ-injective module” and “Σ-essential extension” for unital left modules. Letting Σ be an idempotent topologizing set (called a σ-set below) Σanderson proved the existence of a “Σ-injective hull” for any unital left module and constructed an Utumi Σ-quotient ring of R as the bicommutant of the Σ-injective hull of RR. In this paper, we extend the concepts of “Σinjective module”, “Σ-essentialextension”, and “Σ-injective hull” to modules over arbitrary rings. An overring Σ of a ring R is a Johnson (Utumi) left Σ-quotient ring of R if RR is Σ-essential (Σ-dense) in RS. The maximal Johnson and Utumi Σ-quotient rings of R are constructed similar to the original method of Johnson, and conditions are given to insure their equality. The maximal Utumi Σquotient ring U of R is shown to be the bicommutant of the Σ-injective hull of RR when R has unity. We also obtain a σ-set UΣ of left ideals of U, generated by Σ, and prove that Uis its own maximal Utumi UΣ-quotient ring. A Σ-singular left ideal ZΣ(R) of R is defined and U is shown to be UΣ-injective when Z Σ(R) = 0. The maximal Utumi Σ-quotient rings of matrix rings and direct products of rings are discussed, and the quotient rings of this paper are compared with these of Gabriel (Bull. Soc. Math. France, 90 (1962), 323–448) and Mewborn (Duke Math. J. 35 (1968), 575–580). Our results reduce to those of Johnson and Utumi when 1 ∊ R and Σ is taken to be the set of all left ideals of R.

Derivations of some classes of matrix rings

2017 ◽  
Vol 16 (02) ◽  
pp. 1750027 ◽  
Author(s):  
Feride Kuzucuoğlu ◽  
Umut Sayın
Keyword(s):  
Associative Ring ◽  
Matrix Ring ◽  
Matrix Rings ◽  
Ideal Formula ◽  
The Matrix

Let [Formula: see text] be the ring of all (lower) niltriangular [Formula: see text] matrices over any associative ring [Formula: see text] with identity and [Formula: see text] be the ring of all [Formula: see text] matrices over an ideal [Formula: see text] of [Formula: see text]. We describe all derivations of the matrix ring [Formula: see text].

scholarly journals Quotient rings of semiprime rings with bounded index

1982 ◽  
Vol 23 (1) ◽  
pp. 53-64 ◽  
Author(s):  
John Hannah
Keyword(s):  
Direct Product ◽  
Polynomial Identity ◽  
Semiprime Ring ◽  
Quotient Ring ◽  
Matrix Rings ◽  
Special Cases ◽  
Semiprime Rings ◽  
Strongly Regular ◽  
Quotient Rings ◽  
Finite Direct Product

We say that a ring R has bounded index if there is a positive integer n such that an = 0 for each nilpotent element a of R. If n is the least such integer we say R has index n. For example, any semiprime right Goldie ring has bounded index, and so does any semiprime ring satisfying a polynomial identity [10, Theorem 10.8.2]. This paper is mainly concerned with the maximal (right) quotient ring Q of a semiprime ring R with bounded index. Several special cases of this situation have already received attention in the literature. If R satisfies a polynomial identity [1], or if every nonzero right ideal of R contains a nonzero idempotent [18] then it is known that Q is a finite direct product of matrix rings over strongly regular self-injective rings, the size of the matrices being bounded by the index of R. On the other hand if R is reduced (that is, has index 1) then Q is a direct product of a strongly regular self-injective ring and a biregular right self-injective ring of type III ([2] and [15]; the terminology is explained in [6]). We prove the following generalization of these results (see Theorems 9 and 11).

scholarly journals On uniform bounds of primeness in matrix rings

2004 ◽  
Vol 76 (2) ◽  
pp. 167-174 ◽  
Author(s):  
Konstantin I. Beidar ◽  
Robert Wisbauer
Keyword(s):  
Positive Integer ◽  
Division Algebra ◽  
Associative Ring ◽  
Matrix Ring ◽  
Maximal Dimension ◽  
Uniform Bounds ◽  
Matrix Rings ◽  
The Matrix ◽  
Rank One

AbstractA subset S of an associative ring R is a uniform insulator for R provided a S b ≠ 0 for any nonzero a, b ∈ R. The ring R is called uniformly strongly prime of bound m if R has uniform insulators and the smallest of those has cardinality m. Here we compute these bounds for matrix rings over fields and obtain refinements of some results of van den Berg in this context.More precisely, for a field F and a positive integer k, let m be the bound of the matrix ring Mk(F), and let n be dimF(V), where V is a subspace of Mk(F) of maximal dimension with respect to not containing rank one matrices. We show that m + n = k2. This implies, for example, that n = k2 − k if and only if there exists a (nonassociative) division algebra over F of dimension k.

Quasi-Baer and biregular generalized matrix rings

2017 ◽  
Vol 16 (04) ◽  
pp. 1750067 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Donald D. Davis
Keyword(s):  
Principal Ideal ◽  
Matrix Ring ◽  
Central Idempotent ◽  
Matrix Rings ◽  
Principal Ideals ◽  
Generalized Matrix ◽  
The Right

Generalized matrix rings are ubiquitous in algebra and have relevant applications to analysis. A ring is quasi-Baer (respectively, right p.q.-Baer) in case the right annihilator of any ideal (respectively, principal ideal) is generated by an idempotent. A ring is called biregular if every principal ideal is generated by a central idempotent. In this paper, we identify the ideals and principal ideals, the annihilators of ideals, and the central and semi-central idempotents of a generalized [Formula: see text] matrix ring. We characterize the generalized matrix rings that are quasi-Baer, right p.q.-Baer, prime, and biregular. We provide examples to illustrate these concepts.

scholarly journals Prime and special ideals in structural matrix rings over a ring without unity

1988 ◽  
Vol 45 (2) ◽  
pp. 220-226 ◽  
Author(s):  
L. Van Wyk
Keyword(s):  
Special Class ◽  
Associative Ring ◽  
Matrix Ring ◽  
Boolean Matrix ◽  
Prime Ideals ◽  
Matrix Rings ◽  
Quasi Order ◽  
Structural Matrix Ring ◽  
Complete Matrix ◽  
Structural Matrix

AbstractA. D. Sands showed that there is a 1–1 correspondence between the prime ideals of an arbitraty associative ring R and the complete matrix ring Mn(R) via P→ Mn(P). A structural matrix ring M(B, R) is the ring of all n × n matrices over R with 0 in the positions where the n × n boolean matrix B, B a quasi-order, has 0. The author characterized the special ideals of M(B, R′), in case R′ has unity, for certain special lasses of rings. In this note results of sands and the author are generalized to structural matrix rings over rings without unity. I t turns out that, although the class of prime simple rings is not a special class, Nagata's M-radical has the same form in structural matrix rings as the special radicals studied by the author.

FI-extending generalized matrix rings

2019 ◽  
Vol 19 (01) ◽  
pp. 2050018
Author(s):  
Gary F. Birkenmeier ◽  
Donald D. Davis
Keyword(s):  
Direct Summand ◽  
Matrix Ring ◽  
Sufficient Conditions ◽  
Matrix Rings ◽  
Generalized Matrix

Recall that a module [Formula: see text] is FI-extending if every fully invariant submodule is essential in a direct summand of [Formula: see text]. Let [Formula: see text] be a generalized matrix ring, where [Formula: see text] and [Formula: see text] are rings and [Formula: see text] and [Formula: see text] are bimodules. In this paper, we investigate necessary and/or sufficient conditions on [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text] to be FI-extending.

On the Structure of Quotient Rings Which are QFX Rings

1975 ◽  
Vol 18 (2) ◽  
pp. 203-207
Author(s):  
Samuel L. Dunn
Keyword(s):  
Finite Number ◽  
Quotient Ring ◽  
Local Rings ◽  
Matrix Rings ◽  
Direct Sum ◽  
Quotient Rings

The object of this paper is to consider the relationships between matrix rings and rings having classical quotient rings which are quasi-Frobenius X (QFX) rings. The main result of this paper is Theorem 12, which shows that if S is a ring with a QFX right classical quotient ring T, then T is isomorphic to a direct sum of a finite number of matrix rings over local rings Ui, while S is almost a direct sum of matrix rings over rings Ci, the Ui being right classical quotient rings of the Ci.

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