Definition from ProofWiki:
An algebra over a ring $(G_R,\oplus)$ is an $R$-module $G_R$ over a commutative ring $R$ with a bilinear mapping $\oplus:G^2→G$.
Context: I want to show that an algebra over a ring with many units has many units, but that seems to require that an algebra over a ring with unity has unity (with respect to $\oplus$). I don't why this should be true, but for some obvious examples it is (e.g. products of rings with unity have unity).
Does an algebra over a ring with unity have unity? why?