Given a commutative $*$-ring $(R,*)$, let $M(R,*)$ be the monoid whose elements are matrices over $R$ of all possible shapes and entries, including those that have $0$ columns or $0$ rows. Let the monoid operation be $\oplus$, denoting direct sum. Let $M \sim K$ be an equivalence relation expressing that there exist unitary matrices $U$ and $V$ such that $UMV = K$. The equivalence relation $\sim$ is therefore unitary equivalence. Has the monoid $M(R,*)/\sim$ ever been studied in the literature?
Note 1: The monoid $M(R,*) / \sim$ is commutative, while $M(R,*)$ is not commutative unless $R = 0$ (the zero ring).
Note 2: For $$(R,*) \in \{(0, \operatorname{id}_{0}), (\mathbb R, \operatorname{id}_\mathbb R), (\mathbb C, a + bi \mapsto a - bi), (\mathbb R[\varepsilon]/(\varepsilon^2),a+b\varepsilon \mapsto a+b\varepsilon), (\mathbb R[\varepsilon]/(\varepsilon^2),a+b\varepsilon \mapsto a-b\varepsilon)\}$$ we have that $M(R,*)/\sim$ is isomorphic to a free commutative monoid. In fact, I don't know of any $M(R, *)/\sim$ which is not isomorphic to a free commutative monoid. When such an isomorphism exists, it effectively generalises the Singular Value Decomposition. For the example $M(0, \operatorname{id}_0) / \sim$, the free commutative monoid has only two generators: the unique possible $0\times 1$ matrix and the unique possible $1 \times 0$ matrix.
