A few facts on integrality

The purpose of this note is to collect some theorems and proofs related to integrality in commutative algebra. The note is subdivided into three parts. Part 1 (Integrality over rings) consists of known facts (Theorems 1, 4, 5) and a generalized exercise from [1] (Corollary 3) with a few minor variations (Theorem 2 and Corollary 6). Part 2 (Integrality over ideal semifiltrations) merges integrality over rings (as considered in Part 1) and integrality over ideals (a less-known but still very useful notion; the book [2] is devoted to it) into one general notion that of integrality over ideal semifiltrations (Definition 9). This notion is very general, yet it can be reduced to the basic notion of integrality over rings by a suitable change of base ring (Theorem 7). This reduction allows to extend some standard properties of integrality over rings to the general case (Theorems 8 and 9). Part 3 (Generalizing to two ideal semifiltrations) continues Part 2, adding one more layer of generality. Its main result is a ”relative” version of Theorem 7 (Theorem 11) and a known fact generalized one more time (Theorem 13). This note is supposed to be self-contained (only linear algebra and basic knowledge about rings, ideals and polynomials is assumed). The proofs are constructive. However, when writing down the proofs I focussed on maximal detail (to ensure correctness) rather than on clarity, so the proofs are probably a pain to read. I think of making a short version of this note with the obvious parts of proofs left out.