Algebra over a commutative ring

An algebra over a commutative ring is a unital ring $A$ together with a unital ring homomorphism $\iota\colon R\to A$ from a commutative ring $R$ such that $\iota(R)$ lies in the center of $A$ (equivalently, $\iota(r)a=a\iota(r)$ for all $r\in R$, $a\in A$).

Equivalently, $A$ is an $R$-module and the multiplication map $A\times A\to A$ is $R$-bilinear, with $1_R$ acting as $1_A$. This framework unifies familiar constructions such as polynomial rings and quotients.

Examples:

• The polynomial ring $R[x]$ is an $R$-algebra via the inclusion $R\hookrightarrow R[x]$.
• For an ideal $I\subseteq R$, the quotient ring $R/I$ is an $R$-algebra via the quotient map, where $I$ is an ideal .
• The matrix ring $M_n(R)$ is an $R$-algebra via scalar matrices $r\mapsto rI_n$.