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( σ , τ ) -derivations on prime near rings

Mohammad Ashraf, Asma Ali, Shakir Ali (2004)

Archivum Mathematicum

There is an increasing body of evidence that prime near-rings with derivations have ring like behavior, indeed, there are several results (see for example [1], [2], [3], [4], [5] and [8]) asserting that the existence of a suitably-constrained derivation on a prime near-ring forces the near-ring to be a ring. It is our purpose to explore further this ring like behaviour. In this paper we generalize some of the results due to Bell and Mason [4] on near-rings admitting a special type of derivation...

A canonical directly infinite ring

Mario Petrich, Pedro V. Silva (2001)

Czechoslovak Mathematical Journal

Let ℕ be the set of nonnegative integers and ℤ the ring of integers. Let ℬ be the ring of N × N matrices over ℤ generated by the following two matrices: one obtained from the identity matrix by shifting the ones one position to the right and the other one position down. This ring plays an important role in the study of directly finite rings. Calculation of invertible and idempotent elements of ℬ yields that the subrings generated by them coincide. This subring is the sum of the ideal ℱ consisting of...

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 direct factor theorem for commutative group algebras

William Ullery (1992)

Commentationes Mathematicae Universitatis Carolinae

Suppose F is a field of characteristic p ≠ 0 and H is a p -primary abelian A -group. It is shown that H is a direct factor of the group of units of the group algebra F H .

A generalization of reflexive rings

Mete Burak Çalcı, Huanyin Chen, Sait Halıcıoğlu (2024)

Mathematica Bohemica

We introduce a class of rings which is a generalization of reflexive rings and J -reversible rings. Let R be a ring with identity and J ( R ) denote the Jacobson radical of R . A ring R is called J -reflexive if for any a , b ∈ R , a R b = 0 implies b R a ⊆ J ( R ) . We give some characterizations of a J -reflexive ring. We prove that some results of reflexive rings can be extended to J -reflexive rings for this general setting. We conclude some relations between J -reflexive rings and some related rings. We investigate some extensions of...

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