Abstract
Let k be an infinite field, I an infinite set, V be a k-Vector-space, and g :kI → V a k-linear map. It is shown that if dimk(V) is not too large (under various hypotheses on card(k)and card(I), if it is finite, respectively less than card(k), respectively less than the continuum), then ker(g)must contain elements (ui)i∈I with all but finitely many components ui nonzero. These results are used to prove that every homomorphism from a direct product Π IAiof not-necessarily associative algebras Ai onto an algebra B, where dimk(B) is not too large (in the same senses) is the sum of a map factoring through the projection Π IAionto the product of finitely many of the Ai, and a map into the ideal {b∈ B |bB= Bb={0}} ⊆ B.Detailed consequences are noted in the case where the Aiare Lie algebras. A version of the above result is also obtained with the field k replaced by a commutative valuation ring. This note resembles in that the two papers obtain similar results on homomorphisms on infinite product algebras; but the methods are different, and the hypotheses under which the methods of one note work are in some ways stronger, in others weaker, than those of the other. Also, in we obtain many consequences from our results, while here we aim for brevity, and after one main result about general algebras, restrict ourselves to a couple of quick consequences for Lie algebras. .
