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In [[commutative algebra]], a '''J-0 ring''' is a [[ring (mathematics)|ring]] <math>R</math> such that the set of regular points, that is, points <math>p</math> of the [[spectrum of a ring|spectrum]] at which the localization <math>R_p</math> 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 set|open subset]], 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]] or [[number theory]] 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]]s are J-2 rings; in fact this is part of the definition of an excellent ring.

All [[Dedekind domain]]s of characteristic 0 and all local [[Noetherian ring]]s of dimension at most 1 are J-2 rings. The family of J-2 rings is closed under taking [[localization of a ring|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]] ''k''[''x''<sub>1</sub>,''x''<sub>2</sub>,...] 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''<sub>''n''</sub>. 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.

== See also ==

* [[Excellent ring]]

==References==

* H. Matsumura, ''Commutative algebra'' {{ISBN|0-8053-7026-9}}, chapter 12.

[[Category:Commutative algebra]]


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