Definition 2.0.1.
For the remainder of the course (except as specified), fix a prime number \(p\text{.}\) Define the following standard categories:
- \(\Set\text{:}\) sets.
- \(\Ab\text{:}\) abelian groups.
- \(\Ring\text{:}\) commutative unital rings.
- \(\Mod_A\text{:}\) modules over \(A\) (where \(A \in \Ring\)).
Let \(\Rad(A)\) denote the Jacobson radical of \(A \in \Ring\text{.}\) For \(I\) an ideal of \(A\text{,}\) we say that \(A\) is \(I\)-local if \(I \subseteq \Rad(A)\text{;}\) if \(I = (f)\text{,}\) we also say that \(A\) is \(f\)-local.