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

Abstract Let k be an infinite field, I an infinite set, V a k -vector-space, and g : k I → V a k -linear map. It is shown that if dim k ( 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 ( u i ) i ∈ I with all but finitely many components u i nonzero. These results are used to prove that every homomorphism from a direct product ∏ I A i of not-necessarily-associative algebras A i onto an algebra B , where dim k ( B ) is not too large (in the same senses) is the sum of a map factoring through the projection of ∏ I A i onto the product of finitely many of the A i , and a map into the ideal { b ∈ B | b B = B b = { 0 } } ⊆ B . Detailed consequences are noted in the case where the A i are Lie algebras. A version of the above result is also obtained with the field k replaced by a commutative valuation ring.