scholarly journals On commutativity of one-sideds-unital rings

1992 ◽  
Vol 15 (4) ◽  
pp. 813-818
Author(s):  
H. A. S. Abujabal ◽  
M. A. Khan
Keyword(s):  
Polynomial Identity ◽  
Unital Ring ◽  
Unital Rings

The following theorem is proved: Letr=r(y)>1,s, andtbe non-negative integers. IfRis a lefts-unital ring satisfies the polynomial identity[xy−xsyrxt,x]=0for everyx,y∈R, thenRis commutative. The commutativity of a rights-unital ring satisfying the polynomial identity[xy−yrxt,x]=0for allx,y∈R, is also proved.

scholarly journals A commutativity theorem for lefts-unital rings

1990 ◽  
Vol 13 (4) ◽  
pp. 769-774
Author(s):  
Hamza A. S. Abujabal
Keyword(s):  
Commutativity Theorem ◽  
Unital Ring ◽  
Commutativity Theorems ◽  
Associative Rings ◽  
Unital Rings

In this paper we generalize some well-known commutativity theorems for associative rings as follows: LetRbe a lefts-unital ring. If there exist nonnegative integersm>1,k≥0, andn≥0such that for anyx,yinR,[xky−xnym,x]=0, thenRis commutative.

scholarly journals Categorial properties of compressed zero-divisor graphs of finite commutative rings

2020 ◽  
pp. 2150069
Author(s):  
Alen Đurić ◽  
Sara Jevđnić ◽  
Nik Stopar
Keyword(s):  
Principal Ideal ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Local Rings ◽  
Unital Ring ◽  
Zero Divisor Graph ◽  
Definition Of ◽  
Unital Rings ◽  
Finite Commutative Rings

By modifying the existing definition of a compressed zero-divisor graph [Formula: see text], we define a compressed zero-divisor graph [Formula: see text] of a finite commutative unital ring [Formula: see text], where the compression is performed by means of the associatedness relation (a refinement of the relation used in the definition of [Formula: see text]). We prove that this is the best possible compression which induces a functor [Formula: see text], and that this functor preserves categorial products (in both directions). We use the structure of [Formula: see text] to characterize important classes of finite commutative unital rings, such as local rings and principal ideal rings.

scholarly journals Commutativity of one sideds-unital rings through a Streb's result

1997 ◽  
Vol 20 (2) ◽  
pp. 267-270
Author(s):  
Murtaza A. Quadri ◽  
V. W. Jacob ◽  
M. Ashraf
Keyword(s):  
Unital Ring ◽  
Commutativity Theorems ◽  
Ring Elements ◽  
Unital Rings

The main theorem proved in the present paper states as follows “Letm,k,nandsbe fixed non-negative integers such thatkandnare not simultaneously equal to1andRbe a left (resp right)s-unital ring satisfying[(xmyk)n−xsy,x]=0(resp[(xmyk)n−yxs,x]=0) ThenRis commutative.” Further commutativity of lefts-unital rings satisfying the conditionxt[xm,y]−yr[x,f(y)]xs=0wheref(t)∈t2Z[t]andm>0,t,randsare fixed non-negative integers, has been investigated Finally, we extend these results to the case when integral exponents in the underlying conditions are no longer fixed, rather they depend on the pair of ring elementsxandyfor their values. These results generalize a number of commutativity theorems established recently.

A CHARACTERIZATION OF THE COMMUTATIVE UNITAL RINGS WITH ONLY FINITELY MANY UNITAL SUBRINGS

2008 ◽  
Vol 07 (05) ◽  
pp. 601-622 ◽  
Author(s):  
DAVID E. DOBBS ◽  
GABRIEL PICAVET ◽  
MARTINE PICAVET-L'HERMITTE
Keyword(s):  
Direct Products ◽  
Unital Ring ◽  
Unital Rings

A (commutative unital) ring is said to have FSP if it has only finitely many unital subrings. The singly generated rings that have FSP have been classified. Thus, a characterization of the rings satisfying FSP is obtained by proving that a ring R has FSP if and only if either R is finite or R = ℤ[t1, …, tn] ⊇ ℤ where ℤ[ti] has FSP for each i = 1, …, n. Also, the following characterization is given for the nontrivial ring direct products Πi ∈ I Ri that have FSP: I is finite, each Ri has FSP, and there is at most one i ∈ I such that Ri has characteristic 0.

Additive decomposition of ideals

2018 ◽  
Vol 17 (05) ◽  
pp. 1850085 ◽  
Author(s):  
Ali Mohammad Karparvar ◽  
Babak Amini ◽  
Afshin Amini ◽  
Habib Sharif
Keyword(s):  
Direct Summand ◽  
Commutative Ring ◽  
Polynomial Identity ◽  
Jacobson Radical ◽  
Affirmative Answer ◽  
Additive Decomposition ◽  
Von Neumann ◽  
Unital Ring ◽  
Von Neumann Regular ◽  
Special Case

In this paper, we investigate decomposition of (one-sided) ideals of a unital ring [Formula: see text] as a sum of two (one-sided) ideals, each being idempotent, nil, nilpotent, T-nilpotent, or a direct summand of [Formula: see text]. Among other characterizations, we prove that in a polynomial identity ring every (one-sided) ideal is a sum of a nil (one-sided) ideal and an idempotent (one-sided) ideal if and only if the Jacobson radical [Formula: see text] of [Formula: see text] is nil and [Formula: see text] is von Neumann regular. As a special case, these conditions for a commutative ring [Formula: see text] are equivalent to [Formula: see text] having zero Krull dimension. While assuming Köthe’s conjecture in several occasions to be true, we also raise a question, the affirmative answer to which leads to the truth of the conjecture.

On some orders in ∗-rings based on the core-EP decomposition

2020 ◽  
pp. 2250010
Author(s):  
Janko Marovt ◽  
Dijana Mosić
Keyword(s):  
Partial Order ◽  
The Core ◽  
Unital Ring ◽  
Rings With Involution ◽  
Invertible Elements ◽  
Unital Rings ◽  
Minus Partial Order

We study certain relations in unital rings with involution that are derived from the core-EP decomposition. The notion of the WG pre-order and the C-E partial order is extended from [Formula: see text], the set of all [Formula: see text] matrices over [Formula: see text], to the set [Formula: see text] of all core-EP invertible elements in an arbitrary unital ring [Formula: see text] with involution. A new partial order is introduced on [Formula: see text] by combining the WG pre-order and the well known minus partial order, and a new characterization of the core-EP pre-order in unital proper ∗-rings is presented. Properties of these relations are investigated and some known results are thus generalized.

scholarly journals A GENERALIZATION OF SOME COMMUTATIVITY THEOREMS FOR RINGS I

1990 ◽  
Vol 21 (3) ◽  
pp. 239-245
Author(s):  
H. A. S. ABUJABAL
Keyword(s):  
Polynomial Identity ◽  
Unital Ring ◽  
Commutativity Theorems

In this paper we generalize some well-known commutativity theorems for rings as follows: Let $m > 1$, and $n$, $k$ be non-negative integers. Let $R$ be an $s$ - unital ring satisfying the polynomial identity $[x^ny- y^mx^k, x]=0$, for all $x,y\in R$. Then $R$ is commutative.

scholarly journals Functional identities on upper triangular matrix rings

2020 ◽  
Vol 18 (1) ◽  
pp. 182-193
Author(s):  
He Yuan ◽  
Liangyun Chen
Keyword(s):  
Triangular Matrix ◽  
Matrix Rings ◽  
Unital Ring ◽  
Functional Identities ◽  
Upper Triangular Matrix ◽  
Free Subset

Abstract Let R be a subset of a unital ring Q such that 0 ∈ R. Let us fix an element t ∈ Q. If R is a (t; d)-free subset of Q, then Tn(R) is a (t′; d)-free subset of Tn(Q), where t′ ∈ Tn(Q), $\begin{array}{} t_{ll}' \end{array} $ = t, l = 1, 2, …, n, for any n ∈ N.

Parallel algorithms for matroid intersection and matroid parity

2015 ◽  
Vol 07 (02) ◽  
pp. 1550019
Author(s):  
Jinyu Huang
Keyword(s):  
Parallel Algorithms ◽  
Polynomial Identity ◽  
Black Box ◽  
Matroid Intersection ◽  
Polynomial Identity Testing ◽  
Identity Testing ◽  
Common Base ◽  
Graphic Matroid ◽  
Matroid Parity ◽  
Nc Algorithms

A maximum linear matroid parity set is called a basic matroid parity set, if its size is the rank of the matroid. We show that determining the existence of a common base (basic matroid parity set) for linear matroid intersection (linear matroid parity) is in NC2, provided that there are polynomial number of common bases (basic matroid parity sets). For graphic matroids, we show that finding a common base for matroid intersection is in NC2, if the number of common bases is polynomial bounded. To our knowledge, these algorithms are the first deterministic NC algorithms for matroid intersection and matroid parity. We also give a new RNC2 algorithm that finds a common base for graphic matroid intersection. We prove that if there is a black-box NC algorithm for Polynomial Identity Testing (PIT), then there is an NC algorithm to determine the existence of a common base (basic matroid parity set) for linear matroid intersection (linear matroid parity).

Invo-fine unital rings

2016 ◽  
Vol 11 ◽  
pp. 841-843 ◽  
Author(s):  
Peter V. Danchev
Keyword(s):  
Unital Rings
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