Definition:Represented by Quadratic Form

Definition
Let $R$ be an integral domain.

Let $M$ be an $R$-module.

Let $q : M \to R$ be a quadratic form.

Let $a\in R$.

Then $q$ represents $a$ (over $R$) there exists $x\in M\setminus\{0\}$ with $q(x) = a$.

Also defined as
Some authors do not require the given preimage of $a$ to be nonzero. The only difference is, with that convention, that every quadratic form represents $0$, whereas such forms are otherwise known as isotropic forms.

Also see

 * Definition:Equivalent Quadratic Forms
 * Definition:Isotropic Quadratic Form