• Every prime ideal is irreducible.^[2] Let ${\displaystyle J}$  and ${\displaystyle K}$  be ideals of a commutative ring ${\displaystyle R}$ , with neither one contained in the other. Then there exist ${\displaystyle a\in J\setminus K}$  and ${\displaystyle b\in K\setminus J}$ , where neither is in ${\displaystyle J\cap K}$  but the product is. This proves that a reducible ideal is not prime. A concrete example of this are the ideals ${\displaystyle 2\mathbb {Z} }$  and ${\displaystyle 3\mathbb {Z} }$  contained in ${\displaystyle \mathbb {Z} }$ . The intersection is ${\displaystyle 6\mathbb {Z} }$ , and ${\displaystyle 6\mathbb {Z} }$  is not a prime ideal.
• Every irreducible ideal of a Noetherian ring is a primary ideal,^[1] and consequently for Noetherian rings an irreducible decomposition is a primary decomposition.^[3]
• Every primary ideal of a principal ideal domain is an irreducible ideal.
• Every irreducible ideal is primal.^[4]

An element of an integral domain is prime if and only if the ideal generated by it is a non-zero prime ideal. 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 ${\displaystyle \mathbb {Z} }$  for the ideal ${\displaystyle 4\mathbb {Z} }$  since it is not the intersection of two strictly greater ideals.

In algebraic geometry, if an ideal ${\displaystyle I}$  of a ring ${\displaystyle R}$  is irreducible, then ${\displaystyle V(I)}$  is an irreducible subset in the Zariski topology on the spectrum ${\displaystyle \operatorname {Spec} R}$ . The converse does not hold; for example the ideal ${\displaystyle (x^{2},xy,y^{2})}$  in ${\displaystyle \mathbb {C} [x,y]}$  defines the irreducible variety consisting of just the origin, but it is not an irreducible ideal as ${\displaystyle (x^{2},xy,y^{2})=(x^{2},y)\cap (x,y^{2})}$ .

• Irreducible module
• Irreducible space
• Laskerian ring