algebra

In this definition, all rings are assumed to be rings with identity and all ring homomorphisms are assumed to be unital.

Let $R$ be a ring. An algebra over $R$ is a ring $A$ together with a ring homomorphism $f:R\to Z(A)$, where $Z(A)$ denotes the center of $A$. A subalgebra of $A$ is a subset of $A$ which is an algebra.

Equivalently, an algebra over a ring $R$ is an $R$–module $A$ which is a ring and satisfies the property

$$r\cdot (x*y)=(r\cdot x)*y=x*(r\cdot y)$$

for all $r\in R$ and all $x,y\in A$. Here $\cdot$ denotes $R$-module multiplication and $*$ denotes ring multiplication in $A$. One passes between the two definitions as follows: given any ring homomorphism $f:R⟶Z(A)$, the scalar multiplication rule

$$r\cdot b:=f(r)*b$$

makes $A$ into an $R$-module in the sense of the second definition. Conversely, if $A$ satisfies the requirements of the second definition, then the function $f:R\to A$ defined by $f(r):=r\cdot 1$ is a ring homomorphism from $R$ into $Z(A)$.