scholarly journals Primitive Rings of Infinite Matrices

1964 ◽  
Vol 14 (1) ◽  
pp. 47-53 ◽  
Author(s):  
A. D. Sands
Keyword(s):  
Jacobson Radical ◽  
Final Section ◽  
Matrix Ring ◽  
Primitive Ideal ◽  
Alternative Proof ◽  
Infinite Matrix ◽  
Infinite Matrices ◽  
Matrix Rings ◽  
Primitive Ideals ◽  
Infinite Case

E. C. Posner (5) has shown that a ring R is primitive if and only if the corresponding matrix ring Mn(R) is primitive. From this result he is able to deduce that the primitive ideals in Mn(R) are precisely those ideals of the form Mn(P), where P is a primitive ideal in R. This affords an alternative proof that the Jacobson radical of Mn(R) is Mn(J), where J is the Jacobson radical of R. But Patterson (3, 4) has shown that this last result does not hold in general for rings of infinite matrices and thus that the above result concerning primitive ideals cannot be extended to the infinite case. Nevertheless in this paper we are able to show that Posner's result on primitive rings does extend to infinite matrix rings. Patterson's result depends on showing that if the Jacobson radical J of R is not right vanishing then a certain matrix with entries from J does not lie in the Jacobson radical of the infinite matrix ring. In the final section of this paper we consider a ring R with this property and exhibit a primitive ideal in the infinite matrix ring, which does not arise, as above, from a primitive ideal in R. Finally the Jacobson radical of this ring is determined.

scholarly journals Complementation of the Jacobson group in a matrix ring

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.

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.

Representation of tiled matrix rings as full matrix rings

1989 ◽  
Vol 105 (1) ◽  
pp. 67-72 ◽  
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).

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 On radicals of infinite matrix rings

1969 ◽  
Vol 16 (3) ◽  
pp. 195-203 ◽  
Author(s):  
A. D. Sands
Keyword(s):  
Infinite Matrix ◽  
Natural Numbers ◽  
Index Set ◽  
Matrix Rings ◽  
Infinite Set

Let R be a ring and I an infinite set. We denote by M(R) the ring of row finite matrices over I with entries in R. The set I will be omitted from the notation, as the same index set will be used throughout the paper. For convenience it will be assumed that the set of natural numbers is a subset of I.

scholarly journals On Primitive Ideals in Graded Rings

2008 ◽  
Vol 51 (3) ◽  
pp. 460-466 ◽  
Author(s):  
Agata Smoktunowicz
Keyword(s):  
Prime Ideal ◽  
Jacobson Radical ◽  
Graded Rings ◽  
Dimensional Algebra ◽  
Infinite Dimensional ◽  
Primitive Ideals ◽  
Nil Ring

AbstractLet R = be a graded nil ring. It is shown that primitive ideals in R are homogeneous. Let A = be a graded non-PI just-infinite dimensional algebra and let I be a prime ideal in A. It is shown that either I = ﹛0﹜ or I = A. Moreover, A is either primitive or Jacobson radical.

On semiregular infinite matrix rings

1999 ◽  
Vol 27 (12) ◽  
pp. 5737-5740 ◽  
Author(s):  
Francisco José Costa-Cano ◽  
Juan Jacobo simón
Keyword(s):  
Infinite Matrix ◽  
Matrix Rings

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].

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