Linear Maps On Ki, And Homomorphic Descriptions Of Infinite Direct Produce Algebras

Abstract

Let k  be an  inï¬nite ï¬eld,  I an inï¬nite set,  V be  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 ï¬nite, respectively less than  card(k),  respectively less than the continuum), then ker(g)must contain elements  (u_i)_i∈I   with all but ï¬nitely many components  u_i  nonzero. These results are used to prove that every homomorphism from a direct product    Π _IA_iof 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 Π _IA_ionto the product of ï¬nitely many of the  A_i,  and a map into the ideal  {b∈ B |bB= Bb₌{0}} ⊆ B.Detailed consequences are noted in the case where the  A_iare Lie algebras. A version of the above result is also obtained with the ï¬eld  k  replaced by a commutative valuation ring. This note resembles in that the two papers obtain similar results on homomorphisms on inï¬nite 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. .