Flashcards: Commutative algebra

In mathematics, and specifically in abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.

In commutative algebra, an element b of a commutative ring B is said to be integral over A, a subring of B, if there are n ≥ 1 and such that That is to say, b is a root of a monic polynomial over A.

In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension.

In commutative algebra, the Auslander–Buchsbaum formula, introduced by Auslander and Buchsbaum (, theorem 3.7), states that if R is a commutative Noetherian local ring and M is a non-zero finitely generated R-module of finite projective dimension, then Here pd stands for the projective dimension of a module, and depth for the depth of a module.

In commutative algebra, an integrally closed domain A is an integral domain whose integral closure in its field of fractions is A itself.

In commutative algebra, the Auslander–Buchsbaum theorem states that regular local rings are unique factorization domains.

In mathematics, dimension theory is the study in terms of commutative algebra of the notion dimension of an algebraic variety (and by extension that of a scheme).

In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, usually with singular points.

In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality.

In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded.

In algebra, a Mori domain, named after Yoshiro Mori by Querré (, ), is an integral domain satisfying the ascending chain condition on integral divisorial ideals.

In algebra, a Hilbert ring or a Jacobson ring is a ring such that every prime ideal is an intersection of primitive ideals.

In algebra, a Cohen ring is a field or a complete discrete valuation ring of mixed characteristic whose maximal ideal is generated by p.