Say R is a ring, and denote $\varphi :R\rightarrow R$ a one-to-one homomorphism. Consider the case where $\varphi$ is not onto.
One example is the mapping $(a_{0}, a_{1}, ...) \mapsto (0, a_{0}, a_{1}, ...)$ on $(S^{\mathbb{N}}, \bigoplus, \bigodot)$, the product ring for some arbitrary ring $(S, +, *)$.
Are there any necessary and sufficient conditions on R, for which any monomorphism $\varphi :R\rightarrow R$ is an automorphism?