$\DeclareMathOperator{\Spec}{Spec}$ Let $X = \Spec A$, $Y = \Spec B$ be affine complex varieties, that is reduced $\mathbb{C}$-schemes of finite type. Equivalently we can say that $A$ and $B$ are reduced finitely generated $\mathbb{C}$-algebras.

There should be a ring-theoretic condition on $\phi \in \operatorname{Hom}_\mathbb{C}(B, A) \cong \operatorname{Hom}_\mathsf{Sch}(X, Y)$ detecting whether the corresponding morphism of schemes is injective. Is there a nice way to phrase this condition? (besides the obvious thing saying that for every maximal ideal $\mathfrak{m} \subset B$ there is at most one maximal ideal $\mathfrak{n} \subset A$ such that $\phi^{-1}(\mathfrak{n}) = \mathfrak{m}$).

My thoughts: This condition has to be a generalization of both surjective morphisms and localizations and is stable under composition. But not every injective morphism of varieties is an immersion, e.g.\ the normalization of the cuspidal cubic $y^2 = x^3$.

$\DeclareMathOperator{\Spec}{Spec}$ Let $X = \Spec A$, $Y = \Spec B$ be affine complex varieties, that is reduced $\mathbb{C}$-schemes of finite type. Equivalently we can say that $A$ and $B$ are reduced finitely generated $\mathbb{C}$-algebras.

There should be a ring-theoretic condition on $\phi \in \operatorname{Hom}_\mathbb{C}(B, A) \cong \operatorname{Hom}_\mathsf{Sch}(X, Y)$ detecting whether the corresponding morphism of schemes is injective. Is there a nice way to phrase this condition? (besides the obvious thing saying that for every maximal ideal $\mathfrak{m} \subset B$ there is at most one maximal ideal $\mathfrak{n} \subset A$ such that $\phi^{-1}(\mathfrak{n}) = \mathfrak{m}$).

My thoughts: This condition has to be a generalization of both surjective morphisms and localizations and is stable under composition. But not every injective morphism of varieties is an immersion, e.g. the normalization of the cuspidal cubic $y^2 = x^3$.

$\DeclareMathOperator{\Spec}{Spec}$ Let $X = \Spec A$, $Y = \Spec B$ be affine complex varieties, that is reduced $\mathbb{C}$-schemes of finite type. Equivalently we can say that $A$ and $B$ are reduced finitely generated $\mathbb{C}$-algebras.

There should be a ring-theoretic condition on $\phi \in \operatorname{Hom}_\mathbb{C}(B, A) \cong \operatorname{Hom}_\mathsf{Sch}(X, Y)$ detecting whether the corresponding morphism of schemes is injective. Is there a nice way to phrase this condition? (besides the obvious thing saying that for every maximal ideal $\mathfrak{m} \subset B$ there is at most one maximal ideal $\mathfrak{n} \subset A$ such that $\phi^{-1}(\mathfrak{n}) = \mathfrak{m}$).

$\DeclareMathOperator{\Spec}{Spec}$ Let $X = \Spec A$, $Y = \Spec B$ be affine complex varieties, that is reduced $\mathbb{C}$-schemes of finite type. Equivalently we can say that $A$ and $B$ are reduced finitely generated $\mathbb{C}$-algebras.

There should be a ring-theoretic condition on $\phi \in \operatorname{Hom}_\mathbb{C}(B, A) \cong \operatorname{Hom}_\mathsf{Sch}(X, Y)$ detecting whether the corresponding morphism of schemes is injective. Is there a nice way to phrase this condition? (besides the obvious thing saying that for every maximal ideal $\mathfrak{m} \subset B$ there is at most one maximal ideal $\mathfrak{n} \subset A$ such that $\phi^{-1}(\mathfrak{n}) = \mathfrak{m}$).

My thoughts: This condition has to be a generalization of both surjective morphisms and localizations and is stable under composition. But not every injective morphism of varieties is an immersion, e.g.\ the normalization of the cuspidal cubic $y^2 = x^3$.