Example

(1) Every commutative ring $A$ can be considered as an (associative and commutative) $A$-algebra. (2) Let $E$ be a pseudo-ring (I, § 8, no. 1). Multiplication on $E$ and the unique $\mathbf{Z}$-module structure on $E$ define on $E$ an associative $\mathbf{Z}$-algebra structure. (3) Let $F$ be a set and $A$ a commutative ring. The set $A^F$ of all mappings of $F$ into $A$, with the product ring structure (I, § 8, no. 10) and the product

A-module structure (II, § 1, no. 5) is an associative and commutative A-algebra.

(4) Let E be an A-algebra; the internal laws $(x, y) \mapsto xy + yx$ and $(x, y) \mapsto xy - yx$ define (with the A-module structure on E) two A-algebra structures on E, which are not in general associative; the first law

is always commutative.