Embedding of a ring into a ring with unity

With your multiplication, what is your proposed unity?

it may be helpful to look on the ring with unity as $$ eR \oplus \mathbb{Z} $$ where $e$ is an idempotent which commutes with all other elements and satisfies $e^2=e$

then every element is of the form $er +m$ and we have:

$$ (er+m)(es+n) = e^2rs+ern+mes +mn = e(rs+nr+ms) + mn $$ the unity is the element $1$