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A [[Commutative ring|commutative ring]] with unit element in which any [[Prime ideal|prime ideal]] is the intersection of the maximal ideals containing it, i.e. a ring any integral quotient ring of which has a zero [[Jacobson radical|Jacobson radical]]. For instance, any [[Artinian ring|Artinian ring]], any ring of integers (in general, any [[Dedekind ring|Dedekind ring]] which is not semi-local) or any [[Absolutely-flat ring|absolutely-flat ring]] is a Jacobson ring. On the contrary, a local non-Artinian ring is not a Jacobson ring.
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A [[Commutative ring|commutative ring]] with unit element in which any [[Prime ideal|prime ideal]] is the intersection of the [[maximal ideal]]s containing it, i.e. a ring any integral quotient ring of which has a zero [[Jacobson radical]]. For instance, any [[Artinian ring]], any ring of integers (in general, any [[Dedekind ring]] which is not semi-local) or any [[absolutely-flat ring]] is a Jacobson ring. On the contrary, a local non-Artinian ring is not a Jacobson ring.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054180/j0541801.png" /> is a Jacobson ring and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054180/j0541802.png" /> is an integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054180/j0541803.png" />-algebra or an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054180/j0541804.png" />-algebra of finite type, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054180/j0541805.png" /> will be a Jacobson ring; in particular, a quotient ring of a Jacobson ring is a Jacobson ring. A ring of polynomials in a finite number of variables over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054180/j0541806.png" /> is a Jacobson ring; if the number of variables is infinite, the answer will depend on relations between the number of variables and the cardinality of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054180/j0541807.png" />. A ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054180/j0541808.png" /> is a Jacobson ring if the space of maximal ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054180/j0541809.png" /> is quasi-homeomorphic to the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054180/j05418010.png" />; this definition leads to the concept of a Jacobson scheme.
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If $A$ is a Jacobson ring and $B$ is an integral $A$-algebra or an $A$-algebra of finite type, then $B$ will be a Jacobson ring; in particular, a quotient ring of a Jacobson ring is a Jacobson ring. A ring of polynomials in a finite number of variables over a field $K$ is a Jacobson ring; if the number of variables is infinite, the answer will depend on relations between the number of variables and the cardinality of the field $K$. A ring $A$ is a Jacobson ring if the space of maximal ideals of $A$ is quasi-homeomorphic to the spectrum $\mathrm{Spec}(A)$; this definition leads to the concept of a Jacobson scheme.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR>
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</table>
  
  
  
 
====Comments====
 
====Comments====
Quite generally, a (non-commutative) ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054180/j05418011.png" /> is a Jacobson ring if every prime ideal is an intersection of primitive ideals or, equivalently, if every prime factor ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054180/j05418012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054180/j05418013.png" /> a prime ideal, has zero Jacobson radical. Here an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054180/j05418014.png" /> is primitive if the quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054180/j05418015.png" /> is primitive.
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Quite generally, a (non-commutative) ring $A$ is a Jacobson ring if every prime ideal is an intersection of [[primitive ideal]]s or, equivalently, if every prime factor ring $A/\mathfrak{p}$, $\mathfrak{p}$ a prime ideal, has zero Jacobson radical. Here an ideal $\mathfrak{q} \subset A$ is primitive if the quotient $A/\mathfrak{q}$ is a [[primitive ring]].
  
An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054180/j05418016.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054180/j05418017.png" /> which is a Jacobson ring and for which moreover every primitive ideal has finite codimension is sometimes called a Hilbert algebra. The theorem that a finitely-generated polynomial-identity algebra (cf. [[PI-algebra|PI-algebra]]) over a field is a Hilbert algebra, is a non-commutative generalization of the Hilbert Nullstellensatz, [[#References|[a1]]], Chapt. V. This notion of a Hilbert algebra should not be confused with the one defined in the article [[Hilbert algebra|Hilbert algebra]], which refers to an algebra provided with an involution and an inner product satisfying certain properties.
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An algebra $A$ over a field $k$ which is a Jacobson ring and for which moreover every primitive ideal has finite codimension is sometimes called a Hilbert algebra. The theorem that a finitely-generated polynomial-identity algebra (cf. [[PI-algebra]]) over a field is a Hilbert algebra, is a non-commutative generalization of the Hilbert [[Nullstellen Satz]], [[#References|[a1]]], Chapt. V. This notion of a Hilbert algebra should not be confused with the one defined in the article [[Hilbert algebra]], which refers to an algebra provided with an involution and an inner product satisfying certain properties.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Procesu,  "Rings with polynomial identities" , M. Dekker  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.C. McConnell,  J.C. Robson,  "Noncommutative Noetherian rings" , Wiley  (1987)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Procesu,  "Rings with polynomial identities" , M. Dekker  (1973)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  J.C. McConnell,  J.C. Robson,  "Noncommutative Noetherian rings" , Wiley  (1987)</TD></TR>
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</table>
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Latest revision as of 20:32, 19 January 2016

A commutative ring with unit element in which any prime ideal is the intersection of the maximal ideals containing it, i.e. a ring any integral quotient ring of which has a zero Jacobson radical. For instance, any Artinian ring, any ring of integers (in general, any Dedekind ring which is not semi-local) or any absolutely-flat ring is a Jacobson ring. On the contrary, a local non-Artinian ring is not a Jacobson ring.

If $A$ is a Jacobson ring and $B$ is an integral $A$-algebra or an $A$-algebra of finite type, then $B$ will be a Jacobson ring; in particular, a quotient ring of a Jacobson ring is a Jacobson ring. A ring of polynomials in a finite number of variables over a field $K$ is a Jacobson ring; if the number of variables is infinite, the answer will depend on relations between the number of variables and the cardinality of the field $K$. A ring $A$ is a Jacobson ring if the space of maximal ideals of $A$ is quasi-homeomorphic to the spectrum $\mathrm{Spec}(A)$; this definition leads to the concept of a Jacobson scheme.

References

[1] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)


Comments

Quite generally, a (non-commutative) ring $A$ is a Jacobson ring if every prime ideal is an intersection of primitive ideals or, equivalently, if every prime factor ring $A/\mathfrak{p}$, $\mathfrak{p}$ a prime ideal, has zero Jacobson radical. Here an ideal $\mathfrak{q} \subset A$ is primitive if the quotient $A/\mathfrak{q}$ is a primitive ring.

An algebra $A$ over a field $k$ which is a Jacobson ring and for which moreover every primitive ideal has finite codimension is sometimes called a Hilbert algebra. The theorem that a finitely-generated polynomial-identity algebra (cf. PI-algebra) over a field is a Hilbert algebra, is a non-commutative generalization of the Hilbert Nullstellen Satz, [a1], Chapt. V. This notion of a Hilbert algebra should not be confused with the one defined in the article Hilbert algebra, which refers to an algebra provided with an involution and an inner product satisfying certain properties.

References

[a1] C. Procesu, "Rings with polynomial identities" , M. Dekker (1973)
[a2] J.C. McConnell, J.C. Robson, "Noncommutative Noetherian rings" , Wiley (1987)
How to Cite This Entry:
Jacobson ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jacobson_ring&oldid=12065
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article