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J-2 ring
In commutative algebra, a J-0 ring is a ring such that the set of regular points of the spectrum contains a non-empty open subset, a J-1 ring is a ring such that the set of regular points of the spectrum is an 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 rings are J-2 rings; in fact this is part of the definition of an excellent ring.
All Dedekind domains of characteristic 0 and all local Noetherian rings 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 ''k''() 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 singular points consists of just the ideal (0) and contains no nonempty open sets.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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