A few facts on integrality * BRIEF

The purpose of this note is to collect some theorems and proofs related to integrality in commutative algebra. The note is subdivided into four 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). Part 4 (Accelerating ideal semifiltrations) generalizes Theorem 11 (and thus also Theorem 7) a bit further by considering a generalization of powers of an ideal. Part 5 (Generalizing a lemma by Lombardi) is about an auxiliary result Lombardi used in [3] to prove Kronecker’s Theorem. We extend this auxiliary result here. This note is supposed to be self-contained (only linear algebra and basic knowledge about rings, ideals and polynomials is assumed). This is an attempt to make the proofs as short as possible while keeping them easy to read. If you are stuck following one of the proofs, you can find a more detailed version in [4]. However, normally the proofs in [4] are over-detailed, making them harder to read than the ones below.