Correction to: “On automorphisms of separable algebras”

Abstract

A. Magid has pointed out to us that Lemma 1.8 of [1] is not correct.In [2], Hochschild proves that in any simple algebra over a field every element is a sum of units.It is an elementary exercise to verify that in a finite direct sum of simple algebras every element is a sum of units if and only if at most one of the simple algebra summands is the field Z/(2) of two elements.We thus have the following correction of Lemma 1.8.LEMMA 1.8'.Let A he a separable algebra over the semi-local ring K, then every element in A is a sum of units if and only if every element in AjRad(A) is a sum of units.The proof of Lemma 1.8' is the same as the proof of Lemma 1:8 which appears in [1].Let Z u) be the localization of the integers at the prime (2), then the ring of integers A over Z ω in ζ)(i/Ϊ7) is a separable Z i2) -algebra with no idempotents but 0 and 1 but A/R&d(A) = Z/(2) 0 Z/(2) so A is not generated by its units.These facts may be found on page 234-36 of [3].It is therefore necessary to modify the definition of regular ring given in paragraph 2 on page 30 of [1] in order that Theorem 2,1 R be correct.If A is a separable, finitely generated, protective iϋ-algebra and the center of A is if then an iϋ-subalgebra B of A is called regular in case B is separable over R, the only idempotents in the center of B(& Brικ K are 0 and 1, and every element in B is a sum of units in B.