prime ideal and product

1. Proposition

Let \(A\) be a commutative ring \(\mathfrak{p} \subseteq A\) an ideal TFAE:

\(\mathfrak{a}\) is a prime ideal
If for arbitrary ideals \(I,J \subseteq A\) \(I \cdot J \subseteq \mathfrak{p}\), then \(I \subseteq \mathfrak{p}\) or \(J \subseteq \mathfrak{p}\)

2. Proof

2.1. 1) \(\implies\) 2)

Suppose w.l.o.g. \(I \not\subseteq \mathfrak{p}\), then there exists an \(i \in I \setminus \mathfrak{p}\) Let \(j \in J\), then by construction \(j \cdot i \in J \cdot I \subseteq \mathfrak{p}\) and therefore \(j \cdot i \in \mathfrak{p}\). Since \(\mathfrak{p}\) is a prime ideal, we conclude that \(j \in \mathfrak{p}\)

2.2. 2) \(\implies\) 1)

Let \(a,b \in A\) such that \(a \cdot b \in \mathfrak{p}\). Then \((a) \cdot (b) = (a \cdot b) \subseteq \mathfrak{p}\) hence we conclude that w.l.o.g. \((a) \subseteq \mathfrak{p}\). This shows \(a \in \mathfrak{p}\)