Vol. 63, No. 2, 1976

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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

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
Received: 13 November 1975
Revised: 11 February 1976
Published: 1 April 1976
Kwangil Koh
Jiang Luh
Mohan S. Putcha