| Version 4 |
Version 3 |
| Let $A$ be a ring. An {\em algebra} over $A$ is a ring $B$ together with a ring homomorphism $f: A \longrightarrow Z(B)$, where $Z(B)$ denotes the center of $B$. |
Let $A$ be a ring. An {\em algebra} over $A$ is a ring $B$ together with a ring homomorphism $f: A \longrightarrow Z(B)$, where $Z(B)$ denotes the center of $B$. |
| Equivalently, an algebra is an $A$--module $B$ which is a ring and satisfies the property |
Equivalently, an algebra is an $A$--module $B$ which is a ring and satisfies the property |
| a\cdot(x*y) = (a\cdot x)*y = x*(a\cdot y), |
a\cdot(x*y) = (a\cdot x)*y = x*(a\cdot y), |
| for all $a \in A$ and all $x,y \in B$. Here $\cdot$ denotes $A$--module multiplication and $*$ denotes ring multiplication in $B$. |
for all $a \in A$ and all $x,y \in B$. Here $\cdot$ denotes $A$--module multiplication and $*$ denotes ring multiplication in $B$. |
