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# Two generalizations of T-nilpotence

journal contribution

posted on 2023-05-16, 11:13 authored by Barry GardnerBarry Gardner, Kelarev, AVA commutative ring R is called T-idempotent if, for every sequence r1,r2,â‹¯ âˆˆ R, there exists a positive integer n and an idempotent e âˆˆ R such that r1 â‹¯rn = e. A commutative ring R is called T-stable if, for every sequence r1,r2,â‹¯ R, there exists a positive integer n such that r1â‹¯rn = r1â‹¯rn+1 = â‹¯. We show that a commutative ring is T-idempotent if and only if it is the direct product of a T-nilpotent ring and a Boolean ring. We prove that a commutative ring is T-stable if and only if it is the direct product of a T-nilpotent ring and a Boolean ring satisfying the descending chain condition for idempotents. We describe all commutative T-idempotent and T-stable semigroup rings. Â© Springer-Verlag 1998.

## History

## Publication title

Algebra Colloquium## Volume

5## Issue

4## Pagination

449-458## ISSN

1005-3867## Department/School

School of Natural Sciences## Publisher

Springer-Verlag## Place of publication

Singapore## Repository Status

- Restricted