Definition 1

Given a module E over the ring A, an element x of E is said to be isotropic if $\Phi(x, x) = 0$. A submodule F of E is said to be 1) isotropic if there exists an element $x \neq 0$ of F orthogonal to F; 2) totally isotropic if the restriction of $\Phi$ to F is zero.

When the module E is endowed with a quadratic form Q, an element of E will be said to be isotropic (resp. a submodule of E to be isotropic, or totally isotropic) if this element is isotropic (resp. if this submodule is isotropic, or totally isotropic) relative to the bilinear form associated with Q.

An isotropic vector is simply a vector orthogonal to itself. To say that a submodule F is isotropic means that $F \cap F^0 \neq \{0\}$, or, equivalently, that the restriction of $\Phi$ to F is degenerate; a non-isotropic submodule G of E is therefore a submodule such that the restriction of $\Phi$ to G is non-degenerate. In order that a submodule F of E be totally isotropic, it is necessary and sufficient that $F \subset F^0$. If F is a totally isotropic submodule of E, the same is true of every submodule $F'$ contained in F. The sum of a family of pairwise orthogonal totally isotropic submodules is a totally isotropic submodule. The set of totally isotropic submodules of E, ordered by inclusion, is obviously inductive; it follows that every totally isotropic submodule is contained in a maximal totally isotropic submodule.