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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 $N \times N$ 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 class of finite rings

R. Raghavendran (1970)

Compositio Mathematica

A direct factor theorem for commutative group algebras

William Ullery (1992)

Commentationes Mathematicae Universitatis Carolinae

Suppose $F$ is a field of characteristic $p \neq 0$ 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 generalized Picard group for prime rings

S. Montgomery (1990)

Banach Center Publications

A note on isomorphic commutative group algebras over certain rings.

Danchev, Peter (2005)

Analele Ştiinţifice ale Universităţii “Ovidius" Constanţa. Seria: Matematică

A note on the isomorphism of modular group algebras of $p$ -mixed Abelian groups with divisible $p$ -components.

Danchev, Peter (2006)

Analele Ştiinţifice ale Universităţii “Ovidius" Constanţa. Seria: Matematică

A Note On Units And Divisors Of Zero In Group Rings

N. J. Groenewald, H. J. Schutte (1982)

Publications de l'Institut Mathématique

A Survey of Rings Generated by Units

Ashish K. Srivastava (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

This article presents a brief survey of the work done on rings generated by their units.

Addendum to “Ring elements as sums of units”

Charles Lanski, Attila Maróti (2013)

Open Mathematics

We give a comment to Theorem 1.1 published in our paper “Ring elements as sums of units” [Cent. Eur. J. Math., 2009, 7(3), 395–399].

Admissible groups, symmetric factor sets, and simple algebras.

Mollin, R.A. (1984)

International Journal of Mathematics and Mathematical Sciences

Algebraic analysis in structures with the Kaplansky-Jacobson property

D. Przeworska-Rolewicz (2005)

Studia Mathematica

In 1950 N. Jacobson proved that if u is an element of a ring with unit such that u has more than one right inverse, then it has infinitely many right inverses. He also mentioned that I. Kaplansky proved this in another way. Recently, K. P. Shum and Y. Q. Gao gave a new (non-constructive) proof of the Kaplansky-Jacobson theorem for monoids admitting a ring structure. We generalize that theorem to monoids without any ring structure and we show the consequences of the generalized Kaplansky-Jacobson...