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 $f\colon R \to Z(A)$ , 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 $f\colon R \longrightarrow Z(A)$ , 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 $f\colon R \to A$ defined by $f(r) := r \cdot 1$ is a ring homomorphism from $R$ into $Z(A)$ .