The global dimension of a ring A (denoted gl dim(A)) is the supremum of the set of projective dimensions of all A-modules. By homological algebra, this is equal to: In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound. ... In abstract algebra, a module is a generalization of a vector space. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ...

• the supremum of the set of projective dimensions of all finite A-modules.
• the supremum of the injective dimensions of all A-modules.
• the projective dimension of the residue field A / m, when A is a commutative Noetherian local ring with maximal ideal m.

As an application to commutative algebra, Serre proved that a commutative Noetherian local ring A is regular if and only if it has finite global dimension, in which case the global dimension is precisely the Krull dimension of A. In mathematics, a module is a finitely-generated module if it has a finite generating set. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. ... Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... 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 exactly the same as its Krull dimension. ... In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull (1899 - 1971), is defined to be the number of strict inclusions in a maximal chain of prime ideals. ...

Commutative Ring Theory, Hideyuki Matsumura, Cambridge studies in advanced mathematics 8