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Difference between revisions of "Factorial ring"

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A ring with unique decomposition into factors. More precisely, a factorial ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f0380901.png" /> is an [[Integral domain|integral domain]] in which one can find a system of irreducible elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f0380902.png" /> such that every non-zero element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f0380903.png" /> admits a unique representation
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A ring with unique decomposition into factors. More precisely, a factorial ring $A$ is an [[Integral domain|integral domain]] in which one can find a system of irreducible elements $P$ such that every non-zero element $a\in A$ admits a unique representation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f0380904.png" /></td> </tr></table>
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$$a=u\prod_{p\in P}p^{n(p)},$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f0380905.png" /> is invertible and the non-negative integral exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f0380906.png" /> are non-zero for only a finite number of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f0380907.png" />. Here an element is called irreducible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f0380908.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f0380909.png" /> implies that either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809010.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809011.png" /> is invertible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809012.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809013.png" /> is not invertible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809014.png" />.
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where $u$ is invertible and the non-negative integral exponents $n(p)$ are non-zero for only a finite number of elements $p\in P$. Here an element is called irreducible in $A$ if $p=uv$ implies that either $u$ or $v$ is invertible in $A$, and $p$ is not invertible in $A$.
  
In a factorial ring there is a highest common divisor and a least common multiple of any two elements. A ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809015.png" /> is factorial if and only if it is a [[Krull ring|Krull ring]] and satisfies one of the following equivalent conditions: 1) every divisorial ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809016.png" /> is principal; 2) every prime ideal of height 1 is principal; and 3) every non-empty family of principal ideals has a maximal element, and the intersection of any two principal ideals is principal. Every principal ideal ring is factorial. A Dedekind ring is factorial only if it is a principal ideal ring. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809017.png" /> is a multiplicative system in a factorial ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809018.png" />, then the ring of fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809019.png" /> is factorial. A Zariski ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809020.png" /> is factorial if its completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809021.png" /> is.
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In a factorial ring there is a highest common divisor and a least common multiple of any two elements. A ring $A$ is factorial if and only if it is a [[Krull ring|Krull ring]] and satisfies one of the following equivalent conditions: 1) every divisorial ideal of $A$ is principal; 2) every prime ideal of height 1 is principal; and 3) every non-empty family of principal ideals has a maximal element, and the intersection of any two principal ideals is principal. Every principal ideal ring is factorial. A Dedekind ring is factorial only if it is a principal ideal ring. If $S$ is a multiplicative system in a factorial ring $A$, then the ring of fractions $S^{-1}A$ is factorial. A Zariski ring $R$ is factorial if its completion $\hat R$ is.
  
 
Subrings and quotient rings of a factorial ring need not be factorial. The ring of polynomials over a factorial ring and the ring of formal power series over a field or over a discretely-normed ring are factorial. But the ring of formal power series over a factorial ring need not be factorial.
 
Subrings and quotient rings of a factorial ring need not be factorial. The ring of polynomials over a factorial ring and the ring of formal power series over a field or over a discretely-normed ring are factorial. But the ring of formal power series over a factorial ring need not be factorial.
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For the notion of height see [[Height of an ideal|Height of an ideal]].
 
For the notion of height see [[Height of an ideal|Height of an ideal]].
  
A Zariski ring is a [[Noetherian ring|Noetherian ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809022.png" /> having an [[Ideal|ideal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809023.png" /> such that every ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809024.png" /> is closed in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809025.png" />-adic topology (cf. [[Adic topology|Adic topology]]). The last condition can be replaced by: Every element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809026.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809027.png" /> is invertible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809028.png" />. A Zariski ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809029.png" /> is complete if it is a complete topological space (in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809030.png" />-adic topology). The completion of a Zariski ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809031.png" /> is the completion of the topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809032.png" /> (in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809033.png" />-adic topology). This completion, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809034.png" />, is a Zariski ring (take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038090/f03809035.png" /> as ideal).
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A Zariski ring is a [[Noetherian ring|Noetherian ring]] $R$ having an [[Ideal|ideal]] $\mathfrak a$ such that every ideal in $R$ is closed in the $\mathfrak a$-adic topology (cf. [[Adic topology|Adic topology]]). The last condition can be replaced by: Every element $r\in R$ for which $1-r\in\mathfrak a$ is invertible in $R$. A Zariski ring $(R, \mathfrak a)$ is complete if it is a complete topological space (in the $\mathfrak a$-adic topology). The completion of a Zariski ring $(R,\mathfrak a)$ is the completion of the topological space $R$ (in the $\mathfrak a$-adic topology). This completion, $\bar R$, is a Zariski ring (take $\mathfrak a\bar R$ as ideal).

Revision as of 17:34, 7 July 2014

A ring with unique decomposition into factors. More precisely, a factorial ring $A$ is an integral domain in which one can find a system of irreducible elements $P$ such that every non-zero element $a\in A$ admits a unique representation

$$a=u\prod_{p\in P}p^{n(p)},$$

where $u$ is invertible and the non-negative integral exponents $n(p)$ are non-zero for only a finite number of elements $p\in P$. Here an element is called irreducible in $A$ if $p=uv$ implies that either $u$ or $v$ is invertible in $A$, and $p$ is not invertible in $A$.

In a factorial ring there is a highest common divisor and a least common multiple of any two elements. A ring $A$ is factorial if and only if it is a Krull ring and satisfies one of the following equivalent conditions: 1) every divisorial ideal of $A$ is principal; 2) every prime ideal of height 1 is principal; and 3) every non-empty family of principal ideals has a maximal element, and the intersection of any two principal ideals is principal. Every principal ideal ring is factorial. A Dedekind ring is factorial only if it is a principal ideal ring. If $S$ is a multiplicative system in a factorial ring $A$, then the ring of fractions $S^{-1}A$ is factorial. A Zariski ring $R$ is factorial if its completion $\hat R$ is.

Subrings and quotient rings of a factorial ring need not be factorial. The ring of polynomials over a factorial ring and the ring of formal power series over a field or over a discretely-normed ring are factorial. But the ring of formal power series over a factorial ring need not be factorial.

An integral domain is factorial if and only if its multiplicative semi-group is Gaussian (see Gauss semi-group), and for this reason factorial rings are also called Gaussian rings or Gauss rings.

References

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


Comments

For the notion of height see Height of an ideal.

A Zariski ring is a Noetherian ring $R$ having an ideal $\mathfrak a$ such that every ideal in $R$ is closed in the $\mathfrak a$-adic topology (cf. Adic topology). The last condition can be replaced by: Every element $r\in R$ for which $1-r\in\mathfrak a$ is invertible in $R$. A Zariski ring $(R, \mathfrak a)$ is complete if it is a complete topological space (in the $\mathfrak a$-adic topology). The completion of a Zariski ring $(R,\mathfrak a)$ is the completion of the topological space $R$ (in the $\mathfrak a$-adic topology). This completion, $\bar R$, is a Zariski ring (take $\mathfrak a\bar R$ as ideal).

How to Cite This Entry:
Factorial ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Factorial_ring&oldid=15956
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article