Let $K$ be a multiplicatively written semigroup (either commutative or not) and $H$ a subsemigroup of $K$. We say that $H$ is divisor-closed (in $K$) if $x \in H$ for all $x, y \in K$ such that $x \mid_K y$ (i.e., $y = uxv$ for some $u, v \in K$) and $y \in H$.

Accordingly, we say that a semigroup $S$ is annular (bear with me, I don't have a better word for the moment) if it embeds as a divisor-closed subsemigroup into the multiplicative monoid of a ring (either commutative or not). So here is my question:

What is known, if anything at all, about the abelian groups $G$ (either finite or not) for which $\mathscr B(G)$ is annular? Here, $\mathscr B(G)$ denotes the monoid of zero-sum sequences over $G$, that is, the submonoid of $\mathscr F(G)$, the free abelian monoid with basis $G$, given by the inverse image of $0_G$ under the canonical (monoid) epimorphism $\mathscr F(G) \to G$.

This is a very special case of a more general question (namely, when does a semigroup embeds into a ring etc.?), which, however, I'm afraid is also much harder.

The motivation for the question is the following: Factorization theory (that is, the theory of non-unique factorization) grew up out of algebraic number theory, and a turning point in its history was when the theory, up to that point entirely focused on (commutative, unital) rings, was reforged in the language of monoids, based on the consideration that the latter provide "models" (for studying various phenomena of interest) that wouldn't have been available otherwise, with monoids of zero-sum sequences being probably the most important of these models.