Equi-dimensional ideal

An ideal of an integral domain (finitely generated over some field ) having the following property: All prime ideals associated with the primary ideals from the primary decomposition

have the same dimension, that is, the quotient rings have the same Krull dimension for all . This common dimension is called the dimension of the equi-dimensional ideal .

If is the ring of regular functions on a certain affine variety , then an ideal of it is equi-dimensional if and only if all irreducible components of the subvariety defined by have the same dimension.

Comments

An equi-dimensional ideal is also called an unmixed ideal. Instead of the phrase "dimension of an equi-dimensional ideal" one also uses the term "equi-dimension of an idealequi-dimension" (of the ideal).

An integrally-closed Noetherian domain is an integral domain all principal ideals of which are equi-dimensional, [a1], p. 196.

References

[a1]	O. Zariski, P. Samuel, "Commutative algebra" , 2 , Springer (1975)