Thank you. That's a great answer. I'm working my way through Ravi Vakil's notes, and I'm still on Chapter 5 (well before closed subschemes are defined, let alone quasi-coherent ideal sheaves), but this gives me something to focus on as I come across these concepts.

Yves Diers wrote a book: "Categories of Commutative Algebras" (1992). He defines Zariski categories there, in an attempt to describe categories that behave like commutative algebras do, in an attempt to solve commutative algebra problems using category theory, and to generalize the subject in a way similar to the way that abelian categories have generalized the theory of abelian groups, R-modules, sheaves, homological algebra, etc.