Left Euclidean rings

Hans-Heinrich Brungs

Vol. 45 (1973), No. 1, 27–33

DOI: 10.2140/pjm.1973.45.27

Abstract

It is easy to expand the definition of a commutative Euclidean domain to non commutative rings with zero divisors. Using such a generalized definition it is proved that matrix rings over Euclidean domains are Euclidean, that left principal ideal domains with finitely many maximal left ideals only, which are assumed to be two sided, are Euclidean and that direct sums of Euclidean rings are Euclidean. It follows from this that semi simple rings with d.c.c. are Euclidean.

Mathematical Subject Classification

Primary: 16A14

Milestones

Received: 16 December 1971

Published: 1 March 1973