Article : Homomorphic Descriptions Of Infinite Direct Fabricate Algebras

Title

Homomorphic Descriptions Of Infinite Direct Fabricate Algebras

Author

Y. Anitha Kumari, Dr. D.V. Ramalingareddy

Let J be an  infinite field I an infinite set,  V be  a  J Vector-space, and  g :JI  → V  a    J-linear map.  It is shown that if  dim(V) is not too  large (under various hypotheses on  card(J)and  card(I),  if it is finite, respectively less than  card(J)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     not-necessarily associative  algebras  Ai  onto an  algebra  B,  where  dim(B) is not too  large (in the same senses)  is the sum of a map factoring through the projection ΠIAi onto the product of finately 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 Lie algebras. A version of the above result is also obtained with the field   J replaced by a commutative valuation ring. This note resembles in that the two papers obtain similar results on homeomorphisms on innate product algebras; but the methods are deferent, 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

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