It is well known that every non-unital ring can be embedded into a unital ring (e.g., Dorroh's adjunction). I am curious about the converse: every unital ring can be viewed as a subring of a non-unital ring?
If this converse is not correct, will this be still partially true? I am looking for a non-trivial example. One trivial example would be that $\{0\}$ as a subring of $2\mathbb{Z}$.
PS: by a ring here I mean a set that is an additive abelian group and a multiplicative semigroup, and satisfies the distributive laws.
Take any ring $S$ without identity. If $R$ is any ring with identity, then $R\times S$ does not have identity. Is this what you seek?