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A commutativity theorem for associative rings

Mohammad Ashraf (1995)

Archivum Mathematicum

Let m > 1 , s 1 be fixed positive integers, and let R be a ring with unity 1 in which for every x in R there exist integers p = p ( x ) 0 , q = q ( x ) 0 , n = n ( x ) 0 , r = r ( x ) 0 such that either x p [ x n , y ] x q = x r [ x , y m ] y s or x p [ x n , y ] x q = y s [ x , y m ] x r for all y R . In the present paper it is shown that R is commutative if it satisfies the property Q ( m ) (i.e. for all x , y R , m [ x , y ] = 0 implies [ x , y ] = 0 ).

A subclass of strongly clean rings

Orhan Gurgun, Sait Halicioglu and Burcu Ungor (2015)

Communications in Mathematics

In this paper, we introduce a subclass of strongly clean rings. Let R be a ring with identity, J be the Jacobson radical of R , and let J # denote the set of all elements of R which are nilpotent in R / J . An element a R is called very J # -clean provided that there exists an idempotent e R such that a e = e a and a - e or a + e is an element of J # . A ring R is said to be very J # -clean in case every element in R is very J # -clean. We prove that every very J # -clean ring is strongly π -rad clean and has stable range one. It is shown...

Abelian modules.

Agayev, N., Güngöroğlu, G., Harmanci, A., Halicioğlu, S. (2009)

Acta Mathematica Universitatis Comenianae. New Series

Almost Abelian rings

Junchao Wei (2013)

Communications in Mathematics

A ring R is defined to be left almost Abelian if a e = 0 implies a R e = 0 for a N ( R ) and e E ( R ) , where E ( R ) and N ( R ) stand respectively for the set of idempotents and the set of nilpotents of R . Some characterizations and properties of such rings are included. It follows that if R is a left almost Abelian ring, then R is π -regular if and only if N ( R ) is an ideal of R and R / N ( R ) is regular. Moreover it is proved that (1) R is an Abelian ring if and only if R is a left almost Abelian left idempotent reflexive ring. (2) R is strongly...

An intermediate ring between a polynomial ring and a power series ring

M. Tamer Koşan, Tsiu-Kwen Lee, Yiqiang Zhou (2013)

Colloquium Mathematicae

Let R[x] and R[[x]] respectively denote the ring of polynomials and the ring of power series in one indeterminate x over a ring R. For an ideal I of R, denote by [R;I][x] the following subring of R[[x]]: [R;I][x]: = i 0 r i x i R [ [ x ] ] : ∃ 0 ≤ n∈ ℤ such that r i I , ∀ i ≥ n. The polynomial and power series rings over R are extreme cases where I = 0 or R, but there are ideals I such that neither R[x] nor R[[x]] is isomorphic to [R;I][x]. The results characterizing polynomial rings or power series rings with a certain ring...

Central Armendariz rings.

Agayev, Nazim, Güngöroğlu, Gonca, Harmanci, Abdullah, Halicioğlu, S. (2011)

Bulletin of the Malaysian Mathematical Sciences Society. Second Series

Classification of rings satisfying some constraints on subsets

Moharram A. Khan (2007)

Archivum Mathematicum

Let R be an associative ring with identity 1 and J ( R ) the Jacobson radical of R . Suppose that m 1 is a fixed positive integer and R an m -torsion-free ring with 1 . In the present paper, it is shown that R is commutative if R satisfies both the conditions (i) [ x m , y m ] = 0 for all x , y R J ( R ) and (ii) [ x , [ x , y m ] ] = 0 , for all x , y R J ( R ) . This result is also valid if (ii) is replaced by (ii)’ [ ( y x ) m x m - x m ( x y ) m , x ] = 0 , for all x , y R N ( R ) . Our results generalize many well-known commutativity theorems (cf. [1], [2], [3], [4], [5], [6], [9], [10], [11] and [14]).

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