In algebra, a Hilbert ring or a Jacobson ring is a ring such that every
prime ideal is an intersection of
primitive ideal
In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module. A right primitive ideal is defined similarly. Left and right primitive ideals are always two-sided ideals.
Primitive ideals ar ...
s. For commutative rings primitive ideals are the same as
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
s so in this case a Jacobson ring is one in which every prime ideal is an intersection of maximal ideals.
Jacobson rings were introduced independently by , who named them after
Nathan Jacobson
Nathan Jacobson (October 5, 1910 – December 5, 1999) was an American mathematician.
Biography
Born Nachman Arbiser in Warsaw, Jacobson emigrated to America with his family in 1918. He graduated from the University of Alabama in 1930 and was awar ...
because of their relation to Jacobson radicals, and by , who named them Hilbert rings after
David Hilbert because of their relation to
Hilbert's Nullstellensatz
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ...
.
Jacobson rings and the Nullstellensatz
Hilbert's Nullstellensatz of
algebraic geometry is a special case of the statement that the polynomial ring in finitely many variables over a field is a Hilbert ring. A general form of the Nullstellensatz states that if ''R'' is a Jacobson ring, then so is any finitely generated ''R''-algebra ''S''. Moreover, the pullback of any maximal ideal ''J'' of ''S'' is a maximal ideal ''I'' of ''R'', and ''S/J'' is a finite extension of the field ''R/I''.
In particular a morphism of finite type of Jacobson rings induces a morphism of the maximal spectrums of the rings. This explains why for algebraic varieties over fields it is often sufficient to work with the maximal ideals rather than with all prime ideals, as was done before the introduction of schemes. For more general rings such as local rings, it is no longer true that morphisms of rings induce morphisms of the maximal spectra, and the use of prime ideals rather than maximal ideals gives a cleaner theory.
Examples
*Any field is a Jacobson ring.
*Any principal ideal domain or Dedekind domain with
Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition y ...
zero is a Jacobson ring. In principal ideal domains and Dedekind domains, the nonzero prime ideals are already maximal, so the only thing to check is if the zero ideal is an intersection of maximal ideals. Asking for the Jacobson radical to be zero guarantees this. In principal ideal domains and Dedekind domains, the Jacobson radical vanishes if and only if there are infinitely many prime ideals.
*Any finitely generated algebra over a Jacobson ring is a Jacobson ring. In particular, any finitely generated algebra over a field or the integers, such as the coordinate ring of any affine algebraic set, is a Jacobson ring.
*A local ring has exactly one maximal ideal, so it is a Jacobson ring exactly when that maximal ideal is the only prime ideal. Thus any commutative local ring with
Krull dimension
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally th ...
zero is Jacobson, but if the Krull dimension is 1 or more, the ring cannot be Jacobson.
* showed that any countably generated algebra over an uncountable field is a Jacobson ring.
*
Tate algebra
In algebra, the ring of restricted power series is the subring of a formal power series ring that consists of power series whose coefficients approach zero as degree goes to infinity.. Over a non-archimedean complete field, the ring is also cal ...
s over
non-archimedean field
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields.
The property, typical ...
s are Jacobson rings.
* A commutative ring ''R'' is a Jacobson ring if and only if ''R''
'x'' the ring of polynomials over ''R'', is a Jacobson ring.
[Kaplansky, Theorem 31]
Characterizations
The following conditions on a commutative ring ''R'' are equivalent:
*''R'' is a Jacobson ring
*Every prime ideal of ''R'' is an intersection of maximal ideals.
*Every
radical ideal
In ring theory, a branch of mathematics, the radical of an ideal I of a commutative ring is another ideal defined by the property that an element x is in the radical if and only if some power of x is in I. Taking the radical of an ideal is called ' ...
is an intersection of maximal ideals.
*Every
Goldman ideal In mathematics, a Goldman domain or G-domain is an integral domain ''A'' whose field of fractions is a finitely generated algebra over ''A''.Goldman domains/ideals are called G-domains/ideals in (Kaplansky 1974). They are named after Oscar Goldman. ...
is maximal.
*Every quotient ring of ''R'' by a prime ideal has a zero
Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition y ...
.
*In every quotient ring, the
nilradical is equal to the Jacobson radical.
*Every finitely generated algebra over ''R'' that is a field is finitely generated as an ''R''-module. (
Zariski's lemma In algebra, Zariski's lemma, proved by , states that, if a field is finitely generated as an associative algebra over another field , then is a finite field extension of (that is, it is also finitely generated as a vector space).
An important ...
)
*Every prime ideal ''P'' of ''R'' such that ''R''/''P'' has an element ''x'' with (''R''/''P'')
−1">−1a field is a maximal prime ideal.
*The spectrum of ''R'' is a Jacobson space, meaning that every closed subset is the closure of the set of closed points in it.
*(For Noetherian rings ''R''): ''R'' has no prime ideals ''P'' such that ''R''/''P'' is a 1-dimensional semi-local ring.
Notes
References
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*{{Citation , last=Krull , first=Wolfgang , author-link=Wolfgang Krull , title=Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950 , url=http://ada00.math.uni-bielefeld.de/ICM/ICM1950.2/ , publisher=
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
, location=Providence, R.I. , mr=0045097 , year=1952 , volume=2 , chapter=Jacobsonsches Radikal und Hilbertscher Nullstellensatz , pages=56–64 , access-date=2013-01-03 , archive-url=https://web.archive.org/web/20141129015155/http://ada00.math.uni-bielefeld.de/ICM/ICM1950.2/ , archive-date=2014-11-29 , url-status=dead
Commutative algebra
Ring theory