On subrings of matrix rings over fields

Let K be a commutative field and let V be an n-dimensional vector space over K. We denote by L(V) the ring of all K-linear endomorphisms of V into itself. A subring of L(V) is always assumed to contain the unit element of L(V), but it need not be a vector subspace of the K-algebra L(V). Suppose now that A is a subring of L(V). Then we may consider L(V) as a left or a right A-module. We may also consider V as a left A-module, i.e. if x ∈ V and if f ∈ A then we get the image element f(x) in V. If we choose a basis for V over K then we get the associated matrix representation of L(V). So in this way L(V) is identified with the full matrix ring Mn(K), and in this way the study of subrings of L(V) can be reduced to the study of subrings of Mn(K).

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Integer-valued polynomials over block matrix algebras

Journal of Algebra and Its Applications ◽

10.1142/s021949882050053x ◽

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].

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Representation of tiled matrix rings as full matrix rings

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100001365 ◽

1989 ◽

Vol 105 (1) ◽

pp. 67-72 ◽

Cited By ~ 6

Author(s):

A. W. Chatters

Keyword(s):

Positive Integer ◽

Jacobson Radical ◽

Integral Domain ◽

Matrix Ring ◽

The Other ◽

Matrix Rings ◽

Full Matrix ◽

Image Position ◽

Full Matrix Ring ◽

Unique Maximal Ideal

It can be very difficult to determine whether or not certain rings are really full matrix rings. For example, let p be an odd prime, let H be the ring of quaternions over the integers localized at p, and setThen T is not presented as a full matrix ring, but there is a subring W of H such that T ≅ M2(W). On the other hand, if we take H to be the ring of quaternions over the integers and form T as above, then it is not known whether T ≅ M2(W) for some ring W. The significance of p being an odd prime is that H/pH is a full 2 x 2 matrix ring, whereas H/2H is commutative. Whether or not a tiled matrix ring such as T above can be re-written as a full matrix ring depends on the sizes of the matrices involved in T and H/pH. To be precise, let H be a local integral domain with unique maximal ideal M and suppose that every one-sided ideal of H is principal. Then H/M ≅ Mk(D) for some positive integer k and division ring D. Given a positive integer n. let T be the tiled matrix ring consisting of all n x n matrices with elements of H on and below the diagonal and elements of M above the diagonal. We shall show in Theorem 2.5 that there is a ring W such that T ≅ Mn(W) if and only if n divides k. An important step in the proof is to show that certain idempotents in T/J(T) can be lifted to idempotents in T, where J(T) is the Jacobson radical of T. This technique for lifting idempotents also makes it possible to show that there are (k + n − 1)!/ k!(n−1)! isomorphism types of finitely generated indecomposable projective right T-modules (Theorem 2·10).

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π-Baer rings

Journal of Algebra and Its Applications ◽

10.1142/s0219498818500299 ◽

2018 ◽

Vol 17 (02) ◽

pp. 1850029 ◽

Cited By ~ 4

Author(s):

Gary F. Birkenmeier ◽

Yeliz Kara ◽

Adnan Tercan

Keyword(s):

Left Ideal ◽

Matrix Ring ◽

Idempotent Element ◽

Matrix Rings ◽

Full Matrix ◽

Baer Ring ◽

Baer Rings ◽

Base Ring ◽

Full Matrix Ring ◽

The Right

We say a ring [Formula: see text] is [Formula: see text]-Baer if the right annihilator of every projection invariant left ideal of [Formula: see text] is generated by an idempotent element of [Formula: see text]. In this paper, we study connections between the [Formula: see text]-Baer condition and related conditions such as the Baer, quasi-Baer and [Formula: see text]-extending conditions. The [Formula: see text]-by-[Formula: see text] generalized triangular and the [Formula: see text]-by-[Formula: see text] triangular [Formula: see text]-Baer matrix rings are characterized. Also, we prove that a [Formula: see text]-by-[Formula: see text] full matrix ring over a [Formula: see text]-Baer ring is a [Formula: see text]-Baer ring. In contrast to the Baer condition, it is shown that the [Formula: see text]-Baer condition transfers from a base ring to many of its polynomial extensions. Examples are provided to illustrate and delimit our results.

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Nonsingular rings with essential socles

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700017110 ◽

1974 ◽

Vol 17 (3) ◽

pp. 358-375 ◽

Cited By ~ 1

Author(s):

G. Ivanov

Keyword(s):

Matrix Representation ◽

Matrix Ring ◽

Triangular Matrix ◽

Primitive Idempotent ◽

Matrix Rings ◽

Uniserial Ring ◽

Special Cases ◽

Triangular Matrix Ring ◽

General Method

This paper is a study of nonsingular rings with essential socles. These rings were first investigated by Goldie [5] who studied the Artinian case and showed that an indecomposable nonsingular generalized uniserial ring is isomorphic to a full blocked triangular matrix ring over a sfield. The structure of nonsingular rings in which every ideal generated by a primitive idempotent is uniform was determined for the Artinian case by Gordon [6] and Colby and Rutter [2], and for the semiprimary case by Zaks [12]. Nonsingular rings with essential socles and finite identities were characterized by Gordon [7] and the author [10]. All these results were obtained by representing the rings in question as matrix rings. In this paper a matrix representation of arbitrary nonsingular rings with essential socles is found (section 2). The above results are special cases of this representation. A general method for representing rings as matrices is developed in section 1.

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Semiprime Goldie Generalised Matrix Rings

Canadian Mathematical Bulletin ◽

10.4153/cmb-1995-024-x ◽

1995 ◽

Vol 38 (2) ◽

pp. 174-176 ◽

Cited By ~ 1

Author(s):

Michael V. Clase

Keyword(s):

Matrix Ring ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Matrix Rings ◽

Necessary And Sufficient

AbstractNecessary and sufficient conditions are given for a generalised matrix ring to be semiprime right Goldie.

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Carbonitration of a Tool for Pressing Stainless Steel Pipes

Journal of Engineering Sciences ◽

10.21272/jes.2020.7(2).c3 ◽

2020 ◽

Vol 7 (2) ◽

pp. C17-C21

Author(s):

I. V. Ivanov ◽

M. V. Mohylenets ◽

K. A. Dumenko ◽

L. Kryvchyk ◽

T. S. Khokhlova ◽

...

Keyword(s):

Heat Treatment ◽

Thermal Treatment ◽

Heat Resistance ◽

Alkali Metals ◽

Matrix Ring ◽

High Hardness ◽

Operating Conditions ◽

Matrix Rings ◽

Conventional Heat Treatment ◽

The Stability

To upgrade the operational stability of the tool at LLC “Karbaz”, Sumy, Ukraine, carbonation of tools and samples for research in melts of salts of cyanates and carbonates of alkali metals at 570–580 °C was carried out to obtain a layer thickness of 0.15–0.25 mm and a hardness of 1000–1150 НV. Tests of the tool in real operating conditions were carried out at the press station at LLC “VO Oscar”, Dnipro, Ukraine. The purpose of the test is to evaluate the feasibility of carbonitriding of thermo-strengthened matrix rings and needle-mandrels to improve their stability, hardness, heat resistance, and endurance. If the stability of matrix rings after conventional heat setting varies around 4–6 presses, the rings additionally subjected to chemical-thermal treatment (carbonitration) demonstrated the stability of 7–9 presses due to higher hardness, heat resistance, the formation of a special structure on the surface due to carbonitration in salt melts cyanates and carbonates. Nitrogen and carbon present in the carbonitrided layer slowed down the processes of transformation of solid solutions and coagulation of carbonitride phases. The high hardness of the carbonitrified layer is maintained up to temperatures above 650 °C. If the stability of the needle-mandrels after conventional heat treatment varies between 50–80 presses, the needles, additionally subjected to chemical-thermal treatment (carbonitration) showed the stability of 100–130 presses due to higher hardness, wear resistance, heat resistance, the formation of a special surface structure due to carbonitration in melts of salts of cyanates and carbonates. Keywords: needle-mandrel, matrix ring, pressing, heat treatment, carbonitration.

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Complementation of the Jacobson group in a matrix ring

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700035103 ◽

2005 ◽

Vol 72 (2) ◽

pp. 317-324

Author(s):

David Dolžan

Keyword(s):

Normal Subgroup ◽

Commutative Ring ◽

Jacobson Radical ◽

Matrix Ring ◽

Graded Rings ◽

Matrix Rings ◽

Generators And Relations ◽

Group Of Units ◽

The Matrix ◽

Finite Commutative Ring

The Jacobson group of a ring R (denoted by  = (R)) is the normal subgroup of the group of units of R (denoted by G(R)) obtained by adding 1 to the Jacobson radical of R (J(R)). Coleman and Easdown in 2000 showed that the Jacobson group is complemented in the group of units of any finite commutative ring and also in the group of units a n × n matrix ring over integers modulo ps, when n = 2 and p = 2, 3, but it is not complemented when p ≥ 5. In 2004 Wilcox showed that the answer is positive also for n = 3 and p = 2, and negative in all the remaining cases. In this paper we offer a different proof for Wilcox's results and also generalise the results to a matrix ring over an arbitrary finite commutative ring. We show this by studying the generators and relations that define a matrix ring over a field. We then proceed to examine the complementation of the Jacobson group in the matrix rings over graded rings and prove that complementation depends only on the 0-th grade.

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Strong Regularity in Arbitrary Rings

Nagoya Mathematical Journal ◽

10.1017/s0027763000022984 ◽

1952 ◽

Vol 4 ◽

pp. 51-53 ◽

Cited By ~ 10

Author(s):

Tetsuo Kandô

Keyword(s):

Matrix Ring ◽

Strong Regularity ◽

Full Matrix ◽

Regular Ideal ◽

Strongly Regular ◽

Full Matrix Ring

An element a of a ring R is called regular, if there exists an element x of R such that a×a = a, and a two-sided ideal a in R is said to be regular if each of its elements is regular B. Brown and N. H. McCoy [1] has recently proved that every ring R has a unique maximal regular two-sided ideal M(R), and that M(R) has the following radical-like property: (i) M(R/M(R)) = 0; (ii) if a is a two-sided ideal of R, then M(a) = a ∩ M(R); (iii) M(Rn) = (M(R))n, where Rn denotes a full matrix ring of order n over R. Arens and Kaplansky [2] has defined an element a of R to be strongly regular when there exists an element x of R such that a2x = a. We shall prove in this note that replacing “regularity” by “strong regularity,” we have also a unique maximal strongly regular ideal N(R), and shall investigate some of its properties.

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Derivations of some classes of matrix rings

Journal of Algebra and Its Applications ◽

10.1142/s021949881750027x ◽

2017 ◽

Vol 16 (02) ◽

pp. 1750027 ◽

Cited By ~ 2

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].

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