Vol. 34, No. 1, 1970

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Characterization of separable ideals

Bryce L. Elkins

Vol. 34 (1970), No. 1, 45–49
Abstract

A k-algebra A is called separable if the exact sequence of left Ae = AkA0-modules: 0 J Ae ϕA 0 splits, where p(a b0) = a b; a two-sided ideal A of A is separable in case the k-algebra A∕A is separable.

In this note, we present two characterizations of separable ideals. In particular, one finds that a monic polynomial f k[x] generates a separable ideal if, and only if, f = g1gs, where the gi are monic polynomials which generate pairwise comaximal indecomposable ideals in k[x], and f(a) is a unit in k[a] = k[x]∕f k[x](a = x + f k[x]).

Mathematical Subject Classification
Primary: 16.20
Milestones
Received: 11 November 1969
Published: 1 July 1970
Authors
Bryce L. Elkins