 Irreducible element

A nonzero nonunit $p$ in an integral domain is irreducible when every factorization $p=ab$ has at least one unit factor.

In the context of abstract algebra, particularly in the study of partially ordered sets and rings, an **irreducible element** has a specific definition: 1. **In a Partially Ordered Set**: An element \( x \) in a partially ordered set \( P \) is called irreducible if it cannot be expressed as the meet (greatest lower bound) of two elements from \( P \) unless one of those elements is \( x \) itself.