Let $f:X\to Y$ be a morphism of locally ringed spaces. In this MSE answer, the first definition below is suggested.

Say $f:X\to Y$ is an $R$-immersion of locally ringed spaces if it's a topological embedding and for all $x\in X$, the canonical homomorphism $O_{Y, f(x)}\to O_{X,x}$ is surjective.

Görtz & Wedhorn define $f$ to be an immersion (of schemes, but still) if's a topological embedding with locally closed image and for all $x\in X$, the canonical homomorphism $O_{Y, f(x)}\to O_{X,x}$ is surjective.

Stacks defines $f$ to be an immersion if it's locally closed, i.e a closed immersion followed by an open immersion.

Following the stacks project's definition of immersion (3), the linked answer, this comment, and this comment, make the following assertions.

- An $R$-immersion is an immersion iff it's locally of finite type.
- $R$-immersions are stable under base change.
- An $R$-immersion is an immersion iff its image is locally closed.

**Question.** The proof of the second assertion is said to be 'quite long', but what about the other two? How to prove them?

I ask because in the $C^p$ category, the fact $C^p$ embeddings have locally closed image seems to follow from the local structure of constant rank maps, and for nice schemes I thought this follows from ZMT.