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.

scholarly journals Generalized blocked triangular matrix rings associated with finite abelian centralizer near-rings

1998 ◽  
Vol 41 (1) ◽  
pp. 177-195
Author(s):  
Kirby C. Smith ◽  
Leon Van Wyk
Keyword(s):  
Large Class ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Matrix Rings ◽  
Abelian Centralizer ◽  
Triangular Matrix Ring ◽  
Structural Matrix Ring ◽  
Structural Matrix

For N any member of a large class of finite abelian right centralizer near-rings, the subring of the ring End(N) of endomorphisms of (N, +) generated by the set of right multiplication maps on N is explicitly described as a generalized blocked triangular matrix ring, which in some cases turns out to be a structural matrix ring.

scholarly journals On the small and essential ideals in certain classes of rings

1989 ◽  
Vol 46 (2) ◽  
pp. 262-271 ◽  
Author(s):  
B. W. Green ◽  
L. van Wyk
Keyword(s):  
Commutative Rings ◽  
Matrix Rings ◽  
Small Ideal ◽  
Prufer Domains ◽  
Prüfer Domains ◽  
Complete Matrix ◽  
Structural Matrix

AbstractIt is well known that for a ring with identity the Brown-McCoy radical is the maximal small ideal. However, in certain subrings of complete matrix rings, which we call structural matrix rings, the maximal small and minimal essential ideals coincide.In this paper we characterize a class of commutative and a class of non-commutative rings for which this coincidence occurs, namely quotients of Prüfer domains and structural matrix rings over Brown-McCoy semisimple rings. A similarity between these two classes is obtained.

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 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.

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.

Left Self Distributively Generated Structural Matrix Rings

2005 ◽  
Vol 33 (9) ◽  
pp. 2865-2877
Author(s):  
John A. Lewallen
Keyword(s):  
Matrix Rings ◽  
Structural Matrix

Semiprime Goldie Generalised Matrix Rings

1995 ◽  
Vol 38 (2) ◽  
pp. 174-176 ◽  
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.

scholarly journals Carbonitration of a Tool for Pressing Stainless Steel Pipes

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.

scholarly journals Strongly prime near-rings

1988 ◽  
Vol 31 (3) ◽  
pp. 337-343 ◽  
Author(s):  
N. J. Groenewald
Keyword(s):  
Special Class ◽  
Prime Ideals ◽  
Prime Rings

Strongly prime rings were introduced by Handelman and Lawrence [5] and in [2] Groenewald and Heyman investigated the upper radical determined by the class of all strongly prime rings. In this paper we extend the concept of strongly prime to near-rings. We show that the class M of distributively generated near-rings is a special class in the sense of Kaarli [6]. We also show that if N is any distributively generated near-ring, then UM(N), UM denotes the upper radical determined by the class M, coincides with the intersection of all the strongly prime ideals of N.

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