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. The minimal number of generators of the maximal ideal is always bounded below by the Krull dimension. In symbols, let A be a local ring with maximal ideal m, and suppose that m is generated by a₁,...,a_n. Then in general n ≥ dim A, and A is defined to be regular if and only if n = dim A. In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. ... 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 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. ...

It is equivalent to say that the dimension of the vector space m/m², considered as a vector space over the residue field k=A/m of A, is equal to the dimension of A. See system of parameters.

Regular local rings were originally defined by Wolfgang Krull, but they first became prominent in the work of Oscar Zariski, who showed that geometrically, a regular local ring corresponds to a smooth point on an algebraic variety. Let Y be an algebraic variety contained in affine n-space, and suppose that Y is the vanishing locus of the polynomials f₁,...,f_m. Y is nonsingular at P if Y satisfies a Jacobian condition: If M = (∂f_i/∂x_j) is the matrix of partial derivatives of the defining equations of the variety, then the rank of the matrix found by evaluating M at P is n - dim Y. Zariski proved that Y is nonsingular at P if and only if the local ring of Y at P is regular. This implies that smoothness is an intrinsic property of the variety, in other words it does not depend on where or how the variety is embedded in affine space. It also suggests that regular local rings should have good properties, but before the introduction of techniques from homological algebra very little was known in this direction. Once such techniques were introduced in the 1950s, Auslander and Buchsbaum proved that every regular local ring is a unique factorization domain. Wolfgang Krull (1899 - 1971) was a German mathematician, after whom Krull dimension, the Krull topology, and Krulls principal ideal theorem are named. ... Oscar Zariski was one of the most influential mathematicians working in the field of algebraic geometry in the twentieth century. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... 1950 was a common year starting on Sunday (link will take you to calendar). ... In mathematics, a unique factorization domain (UFD) is, roughly speaking, a commutative ring in which every element can be uniquely written as a product of prime elements, analogous to the fundamental theorem of arithmetic for the integers. ...

Another property suggested by geometric intuition is that the localization of a regular local ring should again be regular. Geometrically, this corresponds to the intuition that if a surface contains a curve, and that curve is smooth, then the surface is smooth near the curve. Again, this lay unsolved until the introduction of homological techniques. However, Jean-Pierre Serre found a homological characterization of regular local rings: A local ring A is regular if and only if A has finite global dimension. It is easy to show that the property of having finite global dimension is preserved under localization, and consequently that localizations of regular local rings at prime ideals are again regular. This allows us to define regularity for all rings, not just local ones: A ring A is said to be regular if its localizations at all of its prime ideals are regular local rings. It is equivalent to say that A has finite global dimension. 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. ...

If A is a regular ring, then it follows that the polynomial ring A[x] and the formal power series ring A[[x]] are both regular. In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. They are also useful to compactly describe sequences and to find closed formulas for recursively defined sequences; this is...

Examples

1. Every field is a regular local ring. These have (Krull) dimension 0. In fact, the fields are exactly the regular local rings of dimension 0.
2. Any discrete valuation ring is a regular local ring of dimension 1 and the regular local rings of dimension 1 are exactly the discrete valuation rings. Specifically, if k is a field and X is an indeterminate, then the ring of formal power series k[[X]] is a regular local ring having (Krull) dimension 1.
3. If p is an ordinary prime number, the ring of p-adic integers is an example of a discrete valuation ring, and consequently a regular local ring, which does not contain a field.
4. More generally, if k is a field and X₁, X₂, ..., X_d are indeterminates, then the ring of formal power series k[[X₁, X₂, ..., X_d]] is a regular local ring having (Krull) dimension d.
5. If Z is the ring of integers and X is an indeterminate, the ring Z[X]_{(2, X)} is an example of a 2-dimensional regular local ring which does not contain a field.