Consequently, an irreducible ideal of a Noetherian ring is primary.
2.
Moreover, while an ideal generated by a prime element is a prime ideal, it is not true in general that an ideal generated by an irreducible element is an irreducible ideal.
3.
This is not true for irreducible ideals : an irreducible ideal may be generated by an element that is not an irreducible element, as is the case in \ mathbb Z for the ideal 4 \ mathbb Z : It is not the intersection of two strictly greater ideals.
4.
This is not true for irreducible ideals : an irreducible ideal may be generated by an element that is not an irreducible element, as is the case in \ mathbb Z for the ideal 4 \ mathbb Z : It is not the intersection of two strictly greater ideals.
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