Definition 1

Let $A$ be a commutative ring. An algebra over $A$ (or an $A$-algebra, or simply an algebra, when no confusion is to be feared) is a set $E$ with a structure defined by giving the following: (1) an $A$-module structure on $E$; (2) an $A$-bilinear mapping (II, § 3, no. 5) of $E \times E$ into $E$.

The $A$-bilinear mapping of $E \times E$ into $E$ occurring in this definition is called the multiplication of the algebra $E$; it is usually denoted by $(x, y) \mapsto x.y$, or simply $(x, y) \mapsto xy$.

Let $(\alpha_i)_{i \in I}$ and $(\beta_j)_{j \in J}$ be two families of elements of $A$, of finite support (I, § 2, no. 1). Then, for all families $(x_i)_{i \in I}$ and $(y_j)_{j \in J}$ of elements of $E$, the general distributivity formula (I, § 3, no. 4)

holds; in particular

The bilinear mapping $(x, y) \mapsto yx$ of $E \times E$ into $E$ and the $A$-module structure on $E$ define an $A$-algebra structure on $E$, called opposite to the given algebra structure. The set $E$ with this new structure is called the opposite algebra to the algebra $E$; it is often denoted by $E^0$. The $A$-algebra $E$ is called commutative if it is identical with its opposite, in other words if multiplication in $E$ is commutative. An isomorphism of $E$ onto $E^0$ is also called an anti-automorphism of the algebra $E$.

When multiplication in the algebra $E$ is associative, $E$ is called an associative $A$-algebra. When multiplication in $E$ admits an identity element (necessarily unique (I, § 2, no. 1)), this element is called the unit element of $E$ and $E$ is called a unital algebra.