Linear maps on kI, and homomorphic images of infinite direct product algebras

Let k be an infinite field, I an infinite set, V 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 ∏IAi of 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 of ∏IAi onto 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 Ai are Lie algebras.A version of the above result is also obtained with the field k replaced by a commutative valuation ring.