Definition 3

A bilinear mapping $\Phi$ of $E \times F$ into $G$ is said to be degenerate on the right (resp. on the left) if there exists a nonzero element $y_0$ of $F$ (resp. $x_0$ of $E$) such that $\Phi(x, y_0) = 0$ for all $x \in E$ (resp. $\Phi(x_0, y) = 0$ for all $y \in F$). We say that $\Phi$ is degenerate if it is degenerate on the right or if it is degenerate on the left.

For $\Phi$ to be nondegenerate on the right (resp. on the left) it is necessary and sufficient that the linear mapping associated on the right (resp. on the left) to $\Phi$ be injective; saying that $\Phi$ is nondegenerate therefore means that the associated linear mappings $d_\Phi$ and $s_\Phi$ are both injective.

Let $(e_i)_{i \in I}$ and $(f_k)_{k \in K}$ be two families of elements of $E$ and $F$, and let $(a_i)_{i \in I}$ and $(b_k)_{k \in K}$ be two families of elements of $A$ and $B$, all but a finite number of which are zero. It follows from equalities (1) to (4), by induction on the number of nonzero coefficients, that we have

If $(e_i)$ and $(f_k)$ are systems of generators of the modules $E$ and $F$, $\Phi$ is therefore completely determined by the elements $g_{ik} = \Phi(e_i, f_k)$. If $(e_i)$ and $(f_k)$ are bases of $E$ and $F$ and if elements $g_{ik}$ of $G$ ($i \in I, k \in K$) are given, then the formula

defines a mapping of $E \times F$ into $G$, which is bilinear and satisfies $\Phi(e_i, f_k) = g_{ik}$. When $(e_i)$ and $(f_k)$ are finite bases, we call $(\Phi(e_i, f_k))$ the matrix of $\Phi$ with respect to these bases.

The bilinear mappings of $E \times F$ into $G$ obviously form a subgroup of the additive group of mappings of $E \times F$ into $G$. On the other hand, let $a$ (resp. $b$) be an element of the center of $A$ (resp. $B$); then the mapping $a \Phi b$ of $E \times F$ into $G$ defined by $(a \Phi b)(x, y) = a . \Phi(x, y) . b$ is bilinear. The set of bilinear mappings of $E \times F$ into $G$ is thus endowed with a bimodule structure over the centers of $A$ and $B$.

Let E' (resp. F') be a left A-module (resp. a right B-module), u (resp. v) a homomorphism of E into E' (resp. of F into F') and $\Phi'$ a bilinear mapping of $E' \times F'$ into G. The inverse image of $\Phi'$ (relative to u and v) is called the bilinear mapping $\Phi$ of $E \times F$ into G defined by

One readily verifies that for all $x \in E,\ y \in F$.

Let $\Phi$ be a bilinear mapping of $E \times F$ into G, and h a homomorphism (for the structures of (A, B)-bimodules) of G into another (A, B)-bimodule $G'$. Then $h \circ \Phi$ is a bilinear mapping of $E \times F$ into $G'$.