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 
, 
,
![\[r(xy) = (rx)y = x(ry) ,\]](http://latex.artofproblemsolving.com/e/a/2/ea27d317813f88117250ccc42d15e2714ea6a444.png)
and
![\[r(x+y) = rx + ry.\]](http://latex.artofproblemsolving.com/e/f/0/ef05a4497c7f89e8fa4cb841a4c207b37d506d0c.png)
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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