# Algebra (structure)

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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