Algebra (structure)

Let $R$ be a commutative ring. We say that a set $E$ is an $R$-algebra if $E$ is an $R$-module and we have a $R$-bilinear mapping of $E\times E$ into $E$, denoted multiplicatively. That is, we have a multiplication between elements of $E$, and between elements of $R$ and elements of $E$ such that for any $r \in R$, $x,y \in E$, \[r(xy) = (rx)y = x(ry) ,\] and \[r(x+y) = rx + ry.\] We identify elements $r$ of $R$ with the corresponding elements $r1$ of $E$.

Note that multiplication in $E$ need not be associative or commutative; however, the elements of $R$ must commute and associate with all elements of $E$. We can thus think of $E$ as an $R$-module endowed with a certain kind of multiplication.

Equivalently, we can say that $E$ is an $R$-algebra if it is a not-necessarily-associative ring that contains $R$ as a sub-ring.

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