When is the tensor product of algebras local? II

Abstract

In a recent paper, Sweedler gave necessary and sufficient conditions for the tensor product of two commutative algebras to be local. We study the noncommutative case. We show that his conditions are necessary in this case. The conditions are not in general, sufficient, and an condition must be replaced by local finiteness. All rings are associative with unity. The Jacobson radical of a ring is denoted by J(R). R is local if A = R/J(R) is a division ring. All algebras and tensor products are over fields. An algebra is if each element generates a dimensional subalgebra. An algebra is if every subset generates a dimensional subalgebra. In a recent paper, Sweedler gave necessary and sufficient conditions for the tensor product of two commutative algebras to be local. THEOREM 1 [4]. Let A and B be commutative F-algebras. Then A OF B is local if and only if: (1) A and B are local, (2) A OF Bf is local, (3) either A or B is F-algebraic. In this paper we show that his conditions are necessary in the noncommutative case. For sufficiency, algebraic must be replaced by locally finite in (3). LEMMA 2. Let A and C be F-algebras such that A ?F C is local. If B is a division subring of A containing F, then B OF C is local. Conversely, if A is a division ring such that for everyfinitely generated division subring B containing F, B ?F C is local, then A OF C is local. PROOF. Suppose a E B 0 C is invertible in A 0 C. Then it is invertible in B 0 C. This can be proved by taking a left B-basis for A. Let {ak} be such a basis with 1 = a, E {a}. Suppose that a = Ei (bi 0 ci) has inverse /3 = Ej (aj c Cj). Then a1 = E blkak, hence 1 = a/i =[ (bi 0 ci)] [E(2 bjkak 0C)] = ( bibjkak 0 cic) By a basis argument, we obtain Received by the editors May 20, 1975, and, in revised form, July 1, 1975. AMS (MOS) subject classifications (1970). Primary 16A48; Secondary 16-00.