Vol. 17, No. 2, 1966

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Simple n-associative rings

David Lewis Outcalt

Vol. 17 (1966), No. 2, 301–309

This paper is concerned with certain classes of nonassociative rings. These rings are defined by first extending the associator (a,b,c) = (ab)c a(bc). The n-associator (a1,,an) is defined by

    (a1,a2) = a1a2,
n∑− 2
(a1,⋅⋅⋅ ,an) =  (− 1)k(a1,⋅⋅⋅ ,ak,ak+1ak+2,⋅⋅⋅ ,an).    (1.1)

A ring is defined to be n-associative if the n-associator vanishes in the ring. It is shown that simple 4-associative and simple 5-associative rings are associative; simple 2k-associative rings are (2k 1) associative or have zero center; and simple, commutative n-associative rings, 6 n 9, are associative. The concept of rings which are associative of degree 2k + 1 is defined, and it is shown that simple, commutative rings which are associative of degree 2k + 1 are associative. The characteristic of the ring is slightly restricted in all but one of these results.

Mathematical Subject Classification
Primary: 17.99
Secondary: 16.96
Received: 19 November 1964
Published: 1 May 1966
David Lewis Outcalt