In this question, it is asked when an abelian group admits a ring structure; that is, if $(G,+)$ is an abelian group, then under what conditions is there a binary operation $\cdot$ on $G$ such that $(G,+,\cdot)$ is a ring. (For the purposes of this question, rings are unital.)
I am interested in the "dual" question of when a monoid admits a ring structure; that is, if $(M,\cdot)$ is a monoid, then under what conditions is there a binary operation $+$ on $M$ such that $(M,+,\cdot)$ is a ring?
The question asks for the description of the image of the forgetful functor $\mathbf{Ring} \to \mathbf{Mon}$ that forgets the additive structure. I don't know if such a description actually exists, but there are some interesting non-trivial necessary conditions for a monoid to be contained in the image.
Well, the first observation is that the underlying multiplicative monoid of a ring is actually a monoid with zero. The functor factors as $\mathbf{Ring} \to \mathbf{Mon}_0$, so that it is more natural to look inside of $\mathbf{Mon}_0$.
There is a basic theorem in ring theory stating that every boolean ring is commutative. This is essentially a statement about the underlying monoid, namely if $x^2=x$ is satisfied for all elements $x$, then $xy=yx$ holds for all elements $x,y$. This does not hold for all monoids: take $\{1,x,y\}$ with $ab := a$ for $a,b \neq 1$, and adjoin a zero if you want a zero.
More generally a ring is called an $n$-ring (for $n > 1$) when every element satisfies $x^n=x$. Jacobson has proven (N. Jacobson, Structure theory for algebraic algebras of bounded degree, Ann. of Math 46.4 (1945), 695–707) that every $n$-ring is commutative, i.e. satisfies $xy=yx$ for all elements $x,y$. Again, this is just a statement about the underlying multiplicative monoids. In this context, there also several theorems of the form that every $n$-ring is an $n'$-ring for some smaller $n'< n$. For example, every $10$-ring is actually a $4$-ring because of the equation $$T^4 - T = (T^2+1) (T^{10}-T) + T^2 ((T+1)^{10}-(T+1))$$ in $\mathbb{F}_2[T]$. So this deduction uses the additive structure, and in fact there are $10$-monoids (with zero) that are not $4$-monoids (with zero): just take the universal example $\langle x : x^{10}=x \rangle$ which consists of $1,x,\dotsc,x^9$ (resp. $0,1,x,\dotsc,x^9$). As a consequence, this cannot come from a ring. You can find more information in my recent preprint Equational proofs of Jacobson's Theorem. The introduction also lists lots of other references around this topic.
Wedderburn's Little Theorem can be seen as a special case of Jacobson's Theorem, it is also a restriction on the monoids with zero associated to a ring. Namely, it states that for every such monoid, in case it is finite and every non-zero element is invertible, is commutative.
If $M$ is the multiplicative monoid of a ring, then $M^{\times}$ is the group of units of that ring - some necessary conditions for groups with this property have been discussed at SE/3367423 and SE/384422.
Maybe it helps to also demand an element $-1 \in M$. Then we define $-x := (-1)x$, and of course we require $(-1)^2 = 1$. This way, the additive inverses are already fixed at least.
