# why do we care about the irreducibility of parameter ideals?

It is well known that a local commutative unital ring $R$ is Gorenstein if and only if every parameter ideal is irreducible. Why the irreducibility of parameter ideals in a Gorenstein local ring is important? I mean I think that it does not seem interesting and useful if the characterization of Gorenstein local rings is the only motivation of dealing with the irreducibility.

• Did you want to ask what does "level" of a reducibility of a parameter ideal mean? – Youngsu Jul 11 '14 at 8:16
• If I don't misunderstand, actually No. I have seen some papers about the "index" of the reducibility of a parameter ideal before. I mean, ok! if $R$ is Gorenstein then every parameter ideal is irreducible. Now what is its application? – Aurora Jul 11 '14 at 8:25