Abstract
Let L be a loop, written multiplicatively, and F an arbitrary field. Define multiplication in the vector space A, of all formal sums of a finite number of elements in L with coefficients in F, by the use of both distributive laws and the definition of multiplication in L. The resulting loop algebra A(L) over F is a linear nonassociative algebra (associative, if and only if L is a group). An algebra A is said to be power associative if the subalgebra FI[x] generated by an element x is an associative algebra for every x of A.
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