Today I have encountered an interesting structure, similar to that of a ring or a semi-ring.
It is a structure $(S, +, \cdot, 1)$, where $S$ is a set, $+, \cdot$ are binary operations, and $1\in S$.
$(S, \cdot, 1)$ is a commutative monoid, $(S, +)$ is a commutative semigroup, and $+$ is distributive with respect to $\cdot$, i. e. $a(b+c) = ab+ac$ for any $a, b, c\in S$.
Does this structure have any names in the literature?
Hopefully, I am not saying anything stupid here.
Consider such a set $S$. Define $R= S \cup \{ 0_R \}$ with the operations extended by $$ 0_R+x =x \\ 0_R\cdot x= 0_R$$
Then $R$ becomes a commutative semi-ring without zero divisors (i.e. $xy=0$ implies $x=0$ or $y=0$).
Converesely, let $R$ be any commutative semi-ring without zero divisors. Then $$S= R \backslash \{ 0 \}$$ satisfies your given conditions.
In other words, your structures are just commutative semi-rings without zero divisors, with the zero removed.