Characterization of separable ideals

Abstract

A -algebra A is called separable if the exact sequence of left A e = A ^-modules: 0 -^> J~> A* -+ A ~+ 0 splits, where (a<S)b) = a-b; a two-sided ideal S of A is separable in case the -algebra A/St is separable. In this note, we present two characterizations of separable ideals. In particular, one finds that a monic polynomial fe k[x] generates a separable ideal if, and only if, / = g 1 g s , where the g are monic polynomials which generate pairwise comaximal indecomposable ideals in k[x], and f(a) is a unit in k[a] = k[x]lf-k[x] (ax + f-k[x\).