Algebra III Study Guide

These flashcards cover key definitions, theorems, and concepts related to projective, injective, flat modules, and Noetherian properties in Algebra.

Projective module

An A-module P is projective if the functor HomA(P, −) is exact.

Injective module

An A-module E is injective if the functor HomA(−, E) is exact.

Flat module

An A-module F is flat if the functor − ⊗A F is exact.

Baer's Criterion

An A-module E is injective if every A-linear homomorphism f : I → E extends to a homomorphism ˜f : A → E for every ideal I ⊆ A.

Noetherian ring

A ring A is Noetherian if every ideal of A is finitely generated.

Noetherian module

An A-module M is Noetherian if every submodule of M is finitely generated.

Tensor product of modules

For A-modules M and N, M ⊗A N is an A-module that satisfies a universal property with bilinear maps.

Primary ideal

An ideal Q ⊊ A is primary if ab ∈ Q and a /∈ Q implies that bn ∈ Q for some n ≥ 1.

Irreducible ideal

An ideal I ⊆ A is irreducible if I = J ∩ K implies I = J or I = K.

Hilbert’s Basis Theorem

If A is a Noetherian ring, then the polynomial ring A[x] is also Noetherian.