commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...

, a J-0 ring is a

ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...

R

such that the set of regular points, that is, points

p

of the

spectrum A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...

at which the localization

R_p

is a regular local ring, contains a non-empty open subset, a J-1 ring is a ring such that the set of regular points is an

open subset In mathematics, an open set is a generalization of an open interval in the real line. In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point in it, contains all points of the met ...

, and a J-2 ring is a ring such that any finitely generated algebra over the ring is a J-1 ring.

Examples

Most rings that occur in

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

are J-2 rings, and in fact it is not trivial to construct any examples of rings that are not. In particular all

excellent ring In commutative algebra, a quasi-excellent ring is a Noetherian ring, Noetherian commutative ring that behaves well with respect to the operation of completion of a ring, completion, and is called an excellent ring if it is also universally catenary ...

s are J-2 rings; in fact this is part of the definition of an excellent ring. All

Dedekind domain In mathematics, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily un ...

s of characteristic 0 and all local

Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals. If the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...

s of dimension at most 1 are J-2 rings. The family of J-2 rings is closed under taking localizations and finitely generated algebras. For an example of a Noetherian domain that is not a J-0 ring, take ''R'' to be the subring of the

polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...

''k'' 'x''₁,''x''₂,...in infinitely many generators generated by the squares and cubes of all generators, and form the ring ''S'' from ''R'' by adjoining inverses to all elements not in any of the ideals generated by some ''x''_''n''. Then ''S'' is a 1-dimensional Noetherian domain that is not a J-0 ring. More precisely ''S'' has a cusp singularity at every closed point, so the set of non-singular points consists of just the ideal (0) and contains no nonempty open sets.

References

* H. Matsumura, ''Commutative algebra'' , chapter 12. Commutative algebra {{commutative-algebra-stub

Examples

See also

References