Equi-dimensional ideal

An ideal $ \mathfrak m $ of an integral domain $ R $( finitely generated over some field $ k $) having the following property: All prime ideals $ \mathfrak P _ {1} \dots \mathfrak P _ {s} $ associated with the primary ideals $ \mathfrak Q _ {1} \dots \mathfrak Q _ {s} $ from the primary decomposition

$$ \mathfrak m = \ \mathfrak Q _ {1} \cap \dots \cap \mathfrak Q _ {s} $$

have the same dimension, that is, the quotient rings $ R/ \mathfrak P _ {i} $ have the same Krull dimension for all $ i $. This common dimension is called the dimension of the equi-dimensional ideal $ \mathfrak m $.

If $ R $ is the ring of regular functions on a certain affine variety $ X $, then an ideal $ \mathfrak m $ of it is equi-dimensional if and only if all irreducible components of the subvariety $ Y \subset X $ defined by $ \mathfrak m $ 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)