Talk:Irreducible ideal

Quick Note

A quick note to fellow editors (and future me). MathWorld states [1]:

"A proper ideal of a ring that is not the intersection of two ideals which properly contain it. In a principal ideal domain, the ideal is irreducible iff [its generator] is an irreducible element."

I don't think this (the second part) is true. For example, a primary ideal of Z is generated by a prime raised to some power and such an element isn't necessarily irreducible. -- Taku (talk) 23:47, 2 April 2009 (UTC)[reply]

I think you are right. Marsupilamov (talk) 20:25, 16 April 2011 (UTC)[reply]

Seems incorrect

This sentence seems incorrect, if I understand it correctly: "An ideal I of a ring A is irreducible if, and only if.the algebraic set it defines is irreducible (that is, any open subset is dense) for the Zariski topology, or equivalently if the closed space of spec A consisting of prime ideals containing I is irreducible for the spectral topology." I believe these statements characterize ideals with prime radical, not irreducible ideals. 173.27.125.112 (talk) 08:51, 26 April 2011 (UTC)[reply]

Source of claim

What is the source of the claim "The converse is not correct, for example the ideal of polynomials in two variables with vanishing terms of first and second order is not irreducible."? -- 139.78.143.6 (talk) 06:08, 31 March 2014 (UTC)[reply]