While working through some lecture notes on semigroups, it seemed to me like a semigroup doesn't buy you much generality over a monoid. But I wondered whether the situation is different for non-unital versus (unital) rings. Then I worked through the following reasons against non-unital rings, which made the point that all naturally occurring non-unital (sub)rings actually have important additional structures, giving ideals and R-algebras as examples. But the author of that paper later later agreed to most of the reasons for non-unital rings.
Let $A$ be a ring (possibly non-unital) and $\tilde{A}=\mathbb Z\oplus A$ as abelian group. I wondered what Martin Brandenburg meant by the "obvious" multiplication so that $A\subseteq \tilde{A}$ is an ideal and $1\in\mathbb Z$ is the identity. After some trying, I came up with $ (r,a)\cdot(s,b)=(rs,rb+sa+ab)$ I haven't checked associativity, but at least the above two conditions are satisfied. Now I wonder what would happen if I replace $\mathbb Z$ by an arbitrary commutative ring $R$ with identity $1$, and assume that I'm also given a scalar multiplication $R \times A \mapsto A$ denoted by $(r, a) \mapsto ra$. Would the above multiplication turn $R\oplus A$ into an (unital) ring, if $A$ is an $R$-algebra? And would conversely $A$ be an $R$-algebra, whenever $R\oplus A$ happens to be an (unital) ring under the above multiplication?