## Linear maps on k^I, and homomorphic images of infinite direct product algebras

##### Authors
Bergman, George M.
Nahlus, Nazih
##### Description
Let k be an infinite field, I an infinite set, V a k-vector-space, and g:k^I\to 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 countable, respectively < card(k)), then ker(g) must contain elements (u_i)_{i\in I} with all but finitely many components u_i nonzero. These results are used to prove that any homomorphism from a direct product \prod_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) must factor through the projection of \prod_I A_i onto the product of finitely many of the A_i, modulo a map into the subalgebra \{b\in B | bB=Bb=\{0\}\}\subseteq B. Detailed consequences are noted in the case where the A_i are Lie algebras.
Comment: 14 pages. Lemma 6 has been strengthened, with resulting strengthening of other results. Some typos etc. have been corrected
##### Keywords
Mathematics - Rings and Algebras, 15A04, 17A01, 17B05 (Primary) 03C20, 03E55, 08B25, 17B20, 17B30 (Secondary)