Algebras Over A Field

Up to now we have not considered the possibility of multiplying two vectors to obtain another vector, though we have noted that this is possible in certain cases. For example, we can multiply elements of the vector space ℳ _n×n (F) over a field F. A vector space V over a field F is an algebra over F if and only if there exists a bilinear transformation (v, w) ↦ vw from V × V to V satisfying the following additional conditions:

for all a ∈ F and all u, v, w ∈ V. In case the additional condition

is satisfied for all u, v, w ∈ V the algebra V is associative. A subspace of V which is closed under this multiplication is called a subalgebra of V.

Authors and Affiliations

1. Department of Mathematics and Computer Science, University of Haifa, Haifa, Israel

   Jonathan S. Golan

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