On the associativity and commutativity of algebras over commutative rings

Kwangil Koh, Jiang Luh and Mohan S. Putcha

Vol. 63 (1976), No. 2, 423–430

DOI: 10.2140/pjm.1976.63.423

Abstract

Let A be an algebra (not necessarily associative) over a commutative ring R. A is left scalar associative if for each a,b,c ∈ A there exists α ∈ R depending on a, b, c such that (ab)c = αa(bc). A right scalar associativity is defined similarly. A is scalar commutative if for each a, b in A, there exists α ∈ R depending on a, b such that αab = ba. In this paper, it is shown that if A is right and left scalar associative and scalar commutative then (ab)c − a(bc) and ab − ba are nilpotent for every a, b and c in A. If 1 ∈ A, then [(ab)c − a(bc)]² = 0. If R is a principal ideal domain then A is associative and commutative.

Mathematical Subject Classification 2000

Primary: 17A30

Milestones

Received: 13 November 1975

Revised: 11 February 1976

Published: 1 April 1976