Irreducible element of a ring

Let \(R\) be a commutative ring. An element \(p\in R\) is irreducible if \(p\) is nonzero, \(p\notin R^\times\), and whenever \(p=ab\) with \(a,b\in R\), \(a \in R^\times\) or \(b\in R^\times\).