scholarly journals 2-Local derivations on associative and Jordan matrix rings over commutative rings

2017 ◽  
Vol 522 ◽  
pp. 28-50 ◽  
Author(s):  
Shavkat Ayupov ◽  
Farhodjon Arzikulov
Keyword(s):  
Commutative Rings ◽  
Matrix Rings ◽  
Jordan Matrix

Some Radical Properties of Jordan Matrix Rings

1979 ◽  
Vol 31 (1) ◽  
pp. 189-196
Author(s):  
Michael Rich
Keyword(s):  
Unique Solution ◽  
Identity Element ◽  
Alternative Ring ◽  
Matrix Rings ◽  
Jordan Ring ◽  
Jordan Matrix ◽  
Image Position ◽  
Canonical Involution

Let A be a ring (not necessarily associative) in which 2x = a has a unique solution for each a ∈ A. Then it is known that if A contains an identity element 1 and an involution j : x ↦ x and if Ja is the canonical involution on An determined by where the ai al−l, 1 ≦ i ≦ n are symmetric elements in the nucleus of A then H(An, Ja), the set of symmetric elements of An, for n ≧ 3 is a Jordan ring if and only if either A is associative or n = 3 and A is an alternative ring whose symmetric elements lie in its nucleus [2, p. 127].

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.

scholarly journals Uniform distribution of sequences in rings of integral matrices

1979 ◽  
Vol 20 (2) ◽  
pp. 169-178
Author(s):  
Harald Niederreiter ◽  
Jau-Shyong Shiue
Keyword(s):  
Uniform Distribution ◽  
Finite Fields ◽  
Commutative Rings ◽  
Algebraic Integers ◽  
Matrix Rings ◽  
Integral Matrices ◽  
Rings Of Algebraic Integers ◽  
Uniform Distribution Of Sequences ◽  
Satisfactory Theory ◽  
Noncommutative Setting

For various discrete commutative rings a concept of uniform distribution has already been introduced and studied, for example, for the ring of rational integers by Niven [9] (see also Kuipers and Niederreiter [2, Ch. 5]), for the rings of Gaussian and Eisenstein integers by Kuipers, Niederreiter, and Shiue [3], for rings of algebraic integers by Lo and Niederreiter [4], [7], and for finite fields by Gotusso [1] and Niederreiter and Shiue [8]. In the present paper, we shall show that a satisfactory theory of uniform distribution can also be developed in a noncommutative setting, namely for matrix rings over the rational integers.

scholarly journals TRACES ON SEMIGROUP RINGS AND LEAVITT PATH ALGEBRAS

2015 ◽  
Vol 58 (1) ◽  
pp. 97-118 ◽  
Author(s):  
ZACHARY MESYAN ◽  
LIA VAŠ
Keyword(s):  
Commutative Rings ◽  
Inverse Semigroups ◽  
Linear Functions ◽  
Group Rings ◽  
Semigroup Rings ◽  
Matrix Rings ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
The Many

AbstractThe trace on matrix rings, along with the augmentation map and Kaplansky trace on group rings, are some of the many examples of linear functions on algebras that vanish on all commutators. We generalize and unify these examples by studying traces on (contracted) semigroup rings over commutative rings. We show that every such ring admits a minimal trace (i.e., one that vanishes only on sums of commutators), classify all minimal traces on these rings, and give applications to various classes of semigroup rings and quotients thereof. We then study traces on Leavitt path algebras (which are quotients of contracted semigroup rings), where we describe all linear traces in terms of central maps on graph inverse semigroups and, under mild assumptions, those Leavitt path algebras that admit faithful traces.

scholarly journals Isomorphism of generalized triangular matrix-rings and recovery of tiles

2003 ◽  
Vol 2003 (9) ◽  
pp. 533-538 ◽  
Author(s):  
R. Khazal ◽  
S. Dăscălescu ◽  
L. Van Wyk
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Commutative Rings ◽  
Local Rings ◽  
Isomorphism Theorem ◽  
Matrix Rings ◽  
Recovery Result

We prove an isomorphism theorem for generalized triangular matrix-rings, over rings having only the idempotents0and1, in particular, over indecomposable commutative rings or over local rings (not necessarily commutative). As a consequence, we obtain a recovery result for the tile in a tiled matrix-ring.

Presentations of skew fields. I. Existentially closed skew fields and the Nullstellensatz

1975 ◽  
Vol 77 (1) ◽  
pp. 7-19 ◽  
Author(s):  
P. M. Cohn
Keyword(s):  
Algebraic Geometry ◽  
Matrix Ring ◽  
Commutative Rings ◽  
Commutative Case ◽  
Matrix Rings ◽  
Finite Dimensional ◽  
Existentially Closed ◽  
Closed Fields ◽  
Skew Fields ◽  
Algebraically Closed Fields

1. Introduction. The Nullstellensatz in commutative algebraic geometry may be described as a means of studying certain commutative rings (viz. affine algebras) by their homomorphisms into algebraically closed fields, and a number of attempts have been made to extend the result to the non-commutative case. In particular, Amitsur and Procesi have studied the case of general rings, with homomorphisms into matrix rings over commutative fields ((1), (2)) and Procesi has obtained more precise results for homomorphisms of PI-rings (11). Since a finite-dimensional division algebra can always be embedded in a matrix ring over a field, this includes the case of skew fields that are finite-dimensional over their centre, but it tells us nothing about general skew fields.

One-Sided Inverses in Rings

1975 ◽  
Vol 27 (1) ◽  
pp. 218-224 ◽  
Author(s):  
Roger D. Peterson
Keyword(s):  
Characteristic Zero ◽  
Commutative Rings ◽  
Group Algebras ◽  
Group Rings ◽  
Matrix Rings ◽  
Von Neumann ◽  
Full Matrix ◽  
Arbitrary Characteristic

Following Herstein [2], we will call a ring R with identity von Neumann finite (vNf) provided that xy = 1 implies yx — WnR. Kaplansky [4] showed that group algebras over fields of characteristic zero are vNf rings, and further, that full matrix rings over such rings are also vNf. Herstein [2] has posed the problem for group algebras over fields of arbitrary characteristic. If group algebras over fields are always vNf, then it is easily seen that group algebras over commutative rings are always vNf. What conditions on the underlying ring of scalars would force the vNf property for all group rings over it?

ON RINGS DETERMINED BY ZERO PRODUCTS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350058 ◽  
Author(s):  
HOGER GHAHRAMANI
Keyword(s):  
Operator Algebras ◽  
Triangular Matrix ◽  
Additive Group ◽  
Commutative Rings ◽  
Matrix Rings ◽  
Full Matrix ◽  
Additive Map ◽  
Upper Triangular Matrix

Let [Formula: see text] be a ring. We say that [Formula: see text] is zero product determined if for every additive group [Formula: see text] and every bi-additive map [Formula: see text] the following holds: if ϕ(a, b) = 0 whenever ab = 0, then there exists an additive map [Formula: see text] such that ϕ(a, b) = T(ab) for all [Formula: see text]. In this paper, first we study the properties of zero product determined rings and show that semi-commutative and non-commutative rings are not zero product determined. Then, we will examine whether the rings with a nontrivial idempotent are zero product determined. As applications of the above results, we prove that simple rings with a nontrivial idempotent, full matrix rings and some classes of operator algebras are zero product determined rings and discuss whether triangular rings and upper triangular matrix rings are zero product determined.

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