Let be a commutative ring. We say that a set is an -algebra if is an -module and we have a -bilinear mapping of into , denoted multiplicatively. That is, we have a multiplication between elements of , and between elements of and elements of such that for any , , and We identify elements of with the corresponding elements of .
Note that multiplication in need not be associative or commutative; however, the elements of must commute and associate with all elements of . We can thus think of as an -module endowed with a certain kind of multiplication.
Equivalently, we can say that is an -algebra if it is a not-necessarily-associative ring that contains as a sub-ring.
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