On strongly right bounded finite rings

1991 ◽  
Vol 44 (3) ◽  
pp. 353-355 ◽  
Author(s):  
Weimin Xue
Keyword(s):  
Left Ideal ◽  
Associative Ring ◽  
Nonzero Ideal ◽  
Finite Ring ◽  
Finite Rings

An associative ring is called strongly right (left) bounded if every nonzero right (left) ideal contains a nonzero ideal. We prove that if R is a strongly right bounded finite ring with unity and the order |R| of R has no factors of the form p5, then R is strongly left bounded. This answers a question of Birkenmeier and Tucci.

scholarly journals ON VARIETIES OF RINGS WHOSE FINITE RINGS ARE DETERMINED BY THEIR ZERO-DIVISOR GRAPHS

2012 ◽  
Vol 05 (02) ◽  
pp. 1250019 ◽  
Author(s):  
A. S. Kuzmina ◽  
Yu. N. Maltsev
Keyword(s):  
Associative Ring ◽  
Zero Divisor ◽  
Finite Ring ◽  
Finite Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors

The zero-divisor graph Γ(R) of an associative ring R is the graph whose vertices are all nonzero zero-divisors (one-sided and two-sided) of R, and two distinct vertices x and y are joined by an edge if and only if either xy = 0 or yx = 0. In the present paper, we study some properties of ring varieties where every finite ring is uniquely determined by its zero-divisor graph.

Stable Rings

1980 ◽  
Vol 23 (2) ◽  
pp. 173-178 ◽  
Author(s):  
S. S. Page
Keyword(s):  
Left Ideal ◽  
Associative Ring ◽  
Commutative Rings ◽  
Von Neumann ◽  
Goldie Dimension ◽  
Von Neumann Regular

Let R be an associative ring with identity. If R is von- Neumann regular of a left v-ring, then for each left ideal, I, we have I2 = I. In this note we study rings such that for each left ideal L there exists an integer n = n(L)>0 such that Ln = Ln+1. We call such rings stable rings. We completely describe the stable commutative rings. These descriptions give rise to concepts related to, but more general than, finite Goldie dimension and T-nilpotence, and a notion of power pure.

scholarly journals THE COMPLEXITY OF THE EQUIVALENCE PROBLEM OVER FINITE RINGS

2011 ◽  
Vol 54 (1) ◽  
pp. 193-199 ◽  
Author(s):  
GÁBOR HORVÁTH
Keyword(s):  
Polynomial Time ◽  
Jacobson Radical ◽  
Equivalence Problem ◽  
Finite Ring ◽  
Polynomial Equations ◽  
Finite Rings

AbstractWe investigate the complexity of the equivalence problem over a finite ring when the input polynomials are written as sum of monomials. We prove that for a finite ring if the factor by the Jacobson radical can be lifted in the centre, then this problem can be solved in polynomial time. This result provides a step in proving a dichotomy conjecture of Lawrence and Willard (J. Lawrence and R. Willard, The complexity of solving polynomial equations over finite rings (manuscript, 1997)).

scholarly journals Weakly prime left ideals in general rings

1985 ◽  
Vol 32 (3) ◽  
pp. 357-360
Author(s):  
Halina France-Jackson
Keyword(s):  
Left Ideal ◽  
Associative Ring ◽  
Almost All ◽  
Associative Rings

A.P.J. van der Walt introduced the concept of a weakly prime left ideal of an associative ring with unity. It is the purpose of the present paper to extend to general, that is not necessarily with unity associative rings, this concept as well as almost all results of van der Walt for rings with unity.

FINITE RINGS WITH EULERIAN ZERO-DIVISOR GRAPHS

2012 ◽  
Vol 11 (03) ◽  
pp. 1250055 ◽  
Author(s):  
A. S. KUZMINA
Keyword(s):  
Associative Ring ◽  
Zero Divisor ◽  
Noncommutative Ring ◽  
Finite Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors

The zero-divisor graph Γ(R) of an associative ring R is the graph whose vertices are all non-zero (one-sided and two-sided) zero-divisors of R, and two distinct vertices x and y are joined by an edge if and only if xy = 0 or yx = 0. [S. P. Redmond, The zero-divisor graph of a noncommutative ring, Int. J. Commut. Rings1(4) (2002) 203–211.] In the present paper, all finite rings with Eulerian zero-divisor graphs are described.

scholarly journals Unit Groups of Classes of Five Radical Zero Commutative Completely Primary Finite Rings

2021 ◽  
pp. 137-154
Author(s):  
Hezron Saka Were ◽  
Maurice Oduor Owino ◽  
Moses Ndiritu Gichuki
Keyword(s):  
Maximal Ideal ◽  
Finite Ring ◽  
Finite Rings ◽  
Zero Divisors ◽  
Unique Maximal Ideal

In this paper, R is considered a completely primary finite ring and Z(R) is its subset of all zero divisors (including zero), forming a unique maximal ideal. We give a construction of R whose subset of zero divisors Z(R) satisfies the conditions (Z(R))5 = (0); (Z(R))4 ̸= (0) and determine the structures of the unit groups of R for all its characteristics.

scholarly journals On n-generalized commutators and Lie ideals of rings

2021 ◽  
pp. 2250221
Author(s):  
Peter V. Danchev ◽  
Tsiu-Kwen Lee
Keyword(s):  
Positive Integer ◽  
Prime Ring ◽  
Classical Result ◽  
Associative Ring ◽  
Nonzero Ideal ◽  
Lie Ideal ◽  
Additive Subgroup ◽  
Simple Ring ◽  
Noncommutative Polynomials ◽  
Generalized Commutator

Let [Formula: see text] be an associative ring. Given a positive integer [Formula: see text], for [Formula: see text] we define [Formula: see text], the [Formula: see text]-generalized commutator of [Formula: see text]. By an [Formula: see text]-generalized Lie ideal of [Formula: see text] (at the [Formula: see text]th position with [Formula: see text]) we mean an additive subgroup [Formula: see text] of [Formula: see text] satisfying [Formula: see text] for all [Formula: see text] and all [Formula: see text], where [Formula: see text]. In the paper, we study [Formula: see text]-generalized commutators of rings and prove that if [Formula: see text] is a noncommutative prime ring and [Formula: see text], then every nonzero [Formula: see text]-generalized Lie ideal of [Formula: see text] contains a nonzero ideal. Therefore, if [Formula: see text] is a noncommutative simple ring, then [Formula: see text]. This extends a classical result due to Herstein [Generalized commutators in rings, Portugal. Math. 13 (1954) 137–139]. Some generalizations and related questions on [Formula: see text]-generalized commutators and their relationship with noncommutative polynomials are also discussed.

Finite Rings in Which 1 is a Sum of Two Non-p-thPower Units

1976 ◽  
Vol 28 (1) ◽  
pp. 94-103 ◽  
Author(s):  
David Jacobson
Keyword(s):  
Prime Number ◽  
Finite Ring ◽  
Finite Rings ◽  
Group Of Units

LetRbe a finite ring with 1 and letR*denote the group of units ofR.Letpbe a prime number. In this paper we consider the question of whether there exista, binR*such thataandb arenon-p-th powers whose sum is 1. If such units a,bexisting, we say that R is an N (p)-ring. Of course ifpdoes not divide |R*|, the order of R*, then every element inR*is apthpower.

Linear Homogeneous Equations Over Finite Rings

1964 ◽  
Vol 16 ◽  
pp. 532-538 ◽  
Author(s):  
Harlan Stevens
Keyword(s):  
Finite Ring ◽  
Finite Rings ◽  
Image Position

The intent of this paper is to apply the following theorem in several particular instances:Theorem 1. For any finite ring of q elements, let {} be a collection of s subsets of , each containing hi (i = 1, 2, . . . , s) members, and let denote the set of all differences d′ — d″ with d′ and d″ from including d′ = d″. Furthermore, suppose that

scholarly journals On Generalized Non-commuting Graph of a Finite Ring

2018 ◽  
Vol 25 (01) ◽  
pp. 149-160 ◽  
Author(s):  
Jutirekha Dutta ◽  
Dhiren K. Basnet ◽  
Rajat K. Nath
Keyword(s):  
Simple Graph ◽  
Finite Ring ◽  
Dominating Sets ◽  
Finite Rings ◽  
Commuting Graph ◽  
Vertex Set

Let S and K be two subrings of a finite ring R. Then the generalized non-commuting graph of subrings S, K of R, denoted by ГS,K, is a simple graph whose vertex set is [Formula: see text], and where two distinct vertices a, b are adjacent if and only if [Formula: see text] or [Formula: see text] and [Formula: see text]. We determine the diameter, girth and some dominating sets for ГS,K. Some connections between ГS,K and Pr(S, K) are also obtained. Further, ℤ-isoclinism between two pairs of finite rings is defined, and we show that the generalized non-commuting graphs of two ℤ-isoclinic pairs are isomorphic under some conditions.

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