commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...

, a J-0 ring is a

ring Ring 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 :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

R

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

p

of the

spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...

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, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...

, and a J-2 ring is a ring such that any

finitely generated algebra In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra ''A'' over a field ''K'' where there exists a finite set of elements ''a''1,...,''a'n'' of ''A'' such that every element of ...

over the ring is a J-1 ring.

Examples

Most rings that occur in

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

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 commutative ring that behaves well with respect to the operation of completion, and is called an excellent ring if it is also universally catenary. Excellent rings are one answer to the ...

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

Dedekind domain In abstract algebra, 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 ...

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-Noether ...

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 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 ...

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 (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) ...

''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