Characterization of separable ideals

Bryce Elkins
1970 Pacific Journal of Mathematics  
A ά-algebra A is called separable if the exact sequence of left A e = A® ^-modules: 0 -^> J~> A* -+ A ~+ 0 splits, where φ(a<S)b°) = a-b; a two-sided ideal Sί of A is separable in case the Λ -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 gι are monic polynomials which generate pairwise comaximal indecomposable ideals in
more » ... osable ideals in k[x], and f(a) is a unit in k[a] = k[x]lf-k
doi:10.2140/pjm.1970.34.45 fatcat:nzqkna2danegdddslq4fsxy4si