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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 .

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