Characterization of separable ideals

Bryce L. Elkins

Vol. 34 (1970), No. 1, 45–49

DOI: 10.2140/pjm.1970.34.45

Abstract

A k-algebra A is called separable if the exact sequence of left A^e = A_⊗kA⁰-modules: 0 → J → A^e → 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 = g₁⋯g_s, where the g_i 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