Definition 1

Given a family $(E_i)_{i \in I}$ of algebras over $A$, the tensor product of this family, denoted by $\bigotimes_{i \in I} E_i$ (or, when $I$ is the interval $\{1, n\}$ of $\mathbf{N}$, $E_1 \otimes_A E_2 \otimes \cdots \otimes_A E_n$, or simply $E_1 \otimes E_2 \otimes \cdots \otimes E_n$) is the algebra obtained by giving the tensor product of the $A$*-modules* $E_i$ the multiplication defined by (1).

Relation (1) shows that the tensor product $\bigotimes_{i \in I} E_i^0$ of the opposite algebras to the $E_i$ is the opposite algebra to $\bigotimes_{i \in I} E_i$; in particular, if the $E_i$ are commutative, so is $\bigotimes_{i \in I} E_i$.

Let $(E_i)_{i \in I}$ and $(F_i)_{i \in I}$ be two families of $A$-algebras with the same finite indexing set $I$. For each $i \in I$, let $f_i : E_i \to F_i$ be an $A$-algebra homomorphism. Then the $A$-linear mapping

is an A-algebra homomorphism, as follows from (1).

For every partition $(I_j)_{j \in J}$ of $I$, the associativity isomorphisms

(II, § 3, no. 9) are also algebra isomorphisms, as follows from (1) and their definitions.

When $I$ is the interval $[1, n]$ of $\mathbf{N}$ and all the algebras $E_i$ are equal to the same algebra $E$, the tensor product algebra $\bigotimes_{i \in I} E_i$ is also denoted by $E^{\otimes n}$.

We shall restrict our attention in the remainder of this no. to the properties of tensor products of two algebras, leaving to the reader the task of extending them to tensor products of arbitrary finite families.

Let $E, F$ be two $A$-algebras; if $a$ (resp. $b$) is a left ideal of $E$ (resp. $F$), the canonical image $\operatorname{Im}(a \otimes b)$ of $a \otimes_A b$ in $E \otimes_A F$ is a left ideal of $E \otimes_A F$; there are analogous results when "left ideal" is replaced by "right ideal" or "two-sided ideal". Moreover: