Let $\varphi$ be one of the following maps.

1. A [group homomorphism](/page/Group%20Homomorphism) $\varphi: G \to H$ between groups. 2. A unital ring homomorphism $\varphi: R \to S$ between unital rings. 3. A [linear map](/page/Linear%20Map) $\varphi: V \to W$ between vector spaces over a fixed field $k$. 4. A [module homomorphism](/page/Module%20Homomorphism) $\varphi: M \to N$ between left modules over a fixed ring $A$.

Then $\operatorname{im}\varphi$ is respectively a subgroup of $H$, a subring of $S$ containing $1_S$, a vector subspace of $W$, or a submodule of $N$.