unmixedness

Unmixedness is a concept studied in commutative algebra and algebraic geometry that describes uniformity of dimension among certain prime ideals associated with a module or a quotient. It is most commonly formulated for Noetherian rings and finitely generated modules.

Definition: Let R be a Noetherian ring and M a finitely generated R-module. The associated primes Ass(M) are primes p that occur as annihilators of elements of M. M is unmixed if all primes p in Ass(M) have the same height, i.e., height(p) is constant on Ass(M). Equivalently, when M ≅ R/I for some ideal I, M is unmixed if all minimal primes over I have the same height; in particular, the quotient R/I has no embedded components of larger height.

Remarks: Unmixedness is weaker than Cohen–Macaulayness; every Cohen–Macaulay module or ring is unmixed, but the converse need not hold. Unmixedness is closely tied to equidimensionality: a ring R is unmixed if and only if R is equidimensional and its localizations avoid embedded associated primes in the sense of primary decomposition.

Examples: If R is a domain and f is a nonzero element, then R/(f) is unmixed of height equal to the height of a prime containing (f). If I is generated by a regular sequence, then R/I is unmixed with height equal to the number of generators. Quotients by ideals with multiple minimal primes of different heights fail to be unmixed.

Significance and variants: Unmixedness captures the geometric notion of an equidimensional subvariety. It appears in the study of primary decomposition, dimension theory, and singularities. In combinatorial commutative algebra, one also speaks of unmixedness for edge ideals of graphs, relating it to well-covered graphs where all minimal vertex covers have the same size.