A general question (sorry for being imprecise and ignorant).
Consider matrices over local ring R. On the set of such matrices acts the left-right group, $GL(n,R)\times GL(m,R)$. The equivalence classes of matrices under this group are modules over $R$. Hence the relation to commutative algebra; various properties of such equivalence classes are translated into the properties of modules.
Suppose we take some subgroup of $G\subset GL(n,R)\times GL(m,R)$ and consider matrices up to $G$-action. The simplest examples: $G=GL(n,R)$ or $G=GL(m,R)$ or (for $m=n$) the congruence. To which objects this corresponds? (In which areas of math such equivalences/objects occur? I guess one should look well beyond commutative algebra.)