Image is a subring

Image is a subring: Let $\varphi:R\to S$ be a ring homomorphism. Then

is a subring of $S$. If $\varphi$ is unital, then $\operatorname{im}(\varphi)$ contains $1_S$.

Thus the image of a ring homomorphism naturally inherits the structure of a subring of the codomain, and in the unital setting it is a unital subring. Combined with the kernel–ideal property , this yields the First Isomorphism Theorem identifying $R/\ker\varphi$ with $\operatorname{im}\varphi$.