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. So here is my question:



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> **Q.** Given an abelian group $G$, denote by $\mathscr B(G)$ 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 $\mathscr F(G) \to G$. For which $G$ is $\mathscr B(G)$ annular?
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This is a very special case of a more general question (namely, when does a semigroup embeds into a ring etc.?), which, however, is also much harder and beyond the scope of this post. Indeed, my motivation is simply 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, until then focused on integral domains, was reforged in the language of monoids, based at least in part 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 representative of these models. So, the whole point of my question is that I'd like to understand to which extent the statement in bold is well-grounded.


*Update #1.* A fruitful exchange with Benjamin Steinberg in the comments below has eventually shown that if $H$ is a non-trivial monoid with trivial group of units (as in the case of interest here) embedding as a divisor-closed subsemigroup into the multiplicative monoid of a unital ring $R$, then the characteristic of $R$ is $2$, simply because the condition of divisor-closedness implies $R^\times \cong H^\times = \{1\_H\}$.



For which abelian groups $G$ does the monoid of zero-sum sequences over $G$ embed into a ring as a divisor-closed subsemigroup?
Figured it out (sorry for answering my own question). I'll prove the following:



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> **Lemma.** Let $H$ be a linearly orderable monoid and $R$ a domain whose group of units is trivial. Then $H$ embeds as a divisor-closed submonoid into the multiplicative monoid of the monoid ring $R[H]$. 
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This will show that the answer to the question in the OP is "For all $G$", since $\mathscr B(G)$ is a torsion-free, cancellative, commutative monoid, and hence is linearly orderable.



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> *Proof of the lemma.* Let $\preceq$ be a total order on $H$ for which $(H, \preceq)$ is a linearly ordered monoid, that is, $xz \prec yz$ and $zx \prec zy$ for all $x, y, z \in H$ with $x \prec y$ (as usual, $u \prec v$ means $u \preceq v$ and $u \ne v$). Moreover, let $\delta$ be the canonical embedding $H \to R[H]$, so that $\delta(x)$ is, for every $x \in H$, a Kronecker delta $H \to R$ centered at $x$, which I'll rather denote by $\delta\_x$.
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> Now pick $x \in H$, and assume $\delta\_x = f\ast g$ for some $f, g \in R[H]$, with $\ast$ being the multiplication in $R[H]$. I claim that $|{\rm supp}(f)| = |{\rm supp}(g)| = 1$. Indeed, it is clear that $f$ and $g$ cannot be identically zero, so the support of each of them is non-empty. Accordingly, let $y\_f$ and $z\_f$ be, respectively, the minimum and the maximum of $S\_f := {\rm supp}(f)$ relative to the order $\preceq$ (which exist because $\preceq$ is total and $S\_f$ is not only non-empty, but also finite); define $y\_g$ and $z\_g$ in a similar way (only with $g$ in lieu of $f$). Then set $y := y\_f y\_g$ and $z := z\_f z\_g$. We have
>  $$ \delta\_x(y) = \sum\_{uv=y} f(u) g(v) = f(y\_f) g(y\_g) \ne 0\_R.$$
>  To see this, note that, if $u \prec y\_f$ or $v \prec y\_g$, then $f(u) g(v) = 0\_R$, and on the other hand, if $y\_f \preceq u$ and $y\_g \preceq v$, then $y = y\_f y\_g \preceq uv$, with equality iff $u = y\_f$ and $v = y\_g$ (by the assumption that $H$ is linearly ordered by $\preceq$). Then the rest is trivial, because $y\_f \in {\rm supp}(f)$ and $y\_g \in {\rm supp}(g)$ give $f(y\_f) g(y\_g) \ne 0\_R$ (by the fact that $R$ is a domain).
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> To wit, we have shown that $y$ is in the support of $\delta\_x$, and so is $z$ by an analogous argument. It follows $x = y = z$, which is, however, possible only if $y\_f = z\_f$ and $y\_g = z\_g$ (by construction, $y\_f \preceq z\_f$ and $y\_g \preceq z\_g$, and we can't have $y\_f \neq z\_f$ or $y\_g \ne z\_g$, otherwise $H$ being linearly ordered by $\preceq$ would yield a contradiction). 
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> In other words, $f = a\delta\_u$ and $g = b \delta\_v$ for some $a, b \in R$ and $u, v \in H$ such that $ab= 1\_R$. But $R^\times$ is trivial (by hypothesis), and therefore $a = b = 1\_R$. So $f, g \in \delta(H)$, and we are done. [] 
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In the last line of the proof, we use that domains are, of course, Dedekind-finite rings, so that a product is a unit iff so are all the factors (see [here](http://mathoverflow.net/questions/261982)).


