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 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.

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 of parameter ideals. Thanks.

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 of parameter ideals.