- is not a unit.
- cannot be written as the product of two non-units in , that is if for some then either or is a unit in .
In a principal ideal domain it is easy to see that the ideal is maximal iff is irreducible. Indeed, we have iff so if is irreducible then or (since , either is a unit (so ) or is times a unit (so )). Conversely if is maximal then if we have so hence either or . In the first case is a unit and in the second case , where is a unit, and hence , a unit. So in either case is irreducible.
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