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''left (right)''
 
''left (right)''
  
A [[Ring|ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n0668501.png" /> satisfying one of the following equivalent conditions:
+
A [[Ring|ring]] $  A $
 +
satisfying one of the following equivalent conditions:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n0668502.png" /> is a left (or right) [[Noetherian module|Noetherian module]] over itself;
+
1) $  A $
 +
is a left (or right) [[Noetherian module|Noetherian module]] over itself;
  
2) every left (or right) ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n0668503.png" /> has a finite generating set;
+
2) every left (or right) ideal in $  A $
 +
has a finite generating set;
  
3) every strictly ascending chain of left (or right) ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n0668504.png" /> breaks off after finitely many terms.
+
3) every strictly ascending chain of left (or right) ideals in $  A $
 +
breaks off after finitely many terms.
  
 
An example of a Noetherian ring is any principal ideal ring, i.e. a ring in which every ideal has one generator.
 
An example of a Noetherian ring is any principal ideal ring, i.e. a ring in which every ideal has one generator.
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Noetherian rings are named after E. Noether, who made a systematic study of such rings and carried over to them a number of results known earlier only under more stringent restrictions (for example, Lasker's theory of primary decompositions).
 
Noetherian rings are named after E. Noether, who made a systematic study of such rings and carried over to them a number of results known earlier only under more stringent restrictions (for example, Lasker's theory of primary decompositions).
  
A right Noetherian ring need not be left Noetherian and vice versa. For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n0668505.png" /> be the ring of matrices of the form
+
A right Noetherian ring need not be left Noetherian and vice versa. For example, let $  A $
 +
be the ring of matrices of the form
  
<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/n/n066/n066850/n0668506.png" /></td> </tr></table>
+
$$
 +
\left \|
 +
\begin{array}{cc}
 +
a  &\alpha  \\
 +
0 &\beta  \\
 +
\end{array}
 +
\right \| ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n0668507.png" /> is a rational integer and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n0668508.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n0668509.png" /> are rational numbers, with the usual addition and multiplication. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n06685010.png" /> is right, but not left, Noetherian, since the left ideal of elements of the form
+
where $  a $
 +
is a rational integer and $  \alpha $
 +
and $  \beta $
 +
are rational numbers, with the usual addition and multiplication. Then $  A $
 +
is right, but not left, Noetherian, since the left ideal of elements of the form
  
<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/n/n066/n066850/n06685011.png" /></td> </tr></table>
+
$$
 +
\left \|
 +
\begin{array}{cc}
 +
0 &\alpha  \\
 +
0  & 0  \\
 +
\end{array}
 +
\right \|
 +
$$
  
 
does not have a finite generating set.
 
does not have a finite generating set.
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Quotient rings and finite direct sums of Noetherian rings are again Noetherian, but a subring of a Noetherian ring need not be Noetherian. For example, a polynomial ring in infinitely many variables over a field is not Noetherian, although it is contained in its field of fractions, which is Noetherian.
 
Quotient rings and finite direct sums of Noetherian rings are again Noetherian, but a subring of a Noetherian ring need not be Noetherian. For example, a polynomial ring in infinitely many variables over a field is not Noetherian, although it is contained in its field of fractions, which is Noetherian.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n06685012.png" /> is a left Noetherian ring, then so is the polynomial ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n06685013.png" />. The corresponding property holds for the ring of formal power series over a Noetherian ring. In particular, polynomial rings of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n06685014.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n06685015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n06685016.png" /> is a field and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n06685017.png" /> the ring of integers, and also quotient rings of them, are Noetherian. Every [[Artinian ring|Artinian ring]] is Noetherian. The localization of a commutative Noetherian ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n06685018.png" /> relative to some multiplicative system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n06685019.png" /> is again Noetherian. If in a commutative Noetherian ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n06685020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n06685021.png" /> is an ideal such that no element of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n06685022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n06685023.png" />, is a divisor of zero, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n06685024.png" />. This means that any such ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n06685025.png" /> defines on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n06685026.png" /> a separable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066850/n06685027.png" />-adic topology. In a commutative Noetherian ring every ideal has a representation as an incontractible intersection of finitely many primary ideals. Although such a representation is not unique, the number of ideals and the set of prime ideals associated with the given primary ideals are uniquely determined.
+
If $  A $
 +
is a left Noetherian ring, then so is the polynomial ring $  A [ X ] $.  
 +
The corresponding property holds for the ring of formal power series over a Noetherian ring. In particular, polynomial rings of the form $  K [ X _ {1} \dots X _ {n} ] $
 +
or $  \mathbf Z [ X _ {1} \dots X _ {n} ] $,  
 +
where $  K $
 +
is a field and $  \mathbf Z $
 +
the ring of integers, and also quotient rings of them, are Noetherian. Every [[Artinian ring|Artinian ring]] is Noetherian. The localization of a commutative Noetherian ring $  A $
 +
relative to some multiplicative system $  S $
 +
is again Noetherian. If in a commutative Noetherian ring $  A $,  
 +
$  \mathfrak m $
 +
is an ideal such that no element of the form $  1 + m $,  
 +
where $  m \in \mathfrak m $,  
 +
is a divisor of zero, then $  \cap _ {k=} 1  ^  \infty  \mathfrak m  ^ {k} = 0 $.  
 +
This means that any such ideal $  \mathfrak m $
 +
defines on $  A $
 +
a separable $  \mathfrak m $-
 +
adic topology. In a commutative Noetherian ring every ideal has a representation as an incontractible intersection of finitely many primary ideals. Although such a representation is not unique, the number of ideals and the set of prime ideals associated with the given primary ideals are uniquely determined.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.L. van der Waerden,  "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1974)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Faith,  "Algebra: rings, modules, and categories" , '''1''' , Springer  (1973)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.L. van der Waerden,  "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1974)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Faith,  "Algebra: rings, modules, and categories" , '''1''' , Springer  (1973)</TD></TR></table>

Latest revision as of 14:54, 7 June 2020


left (right)

A ring $ A $ satisfying one of the following equivalent conditions:

1) $ A $ is a left (or right) Noetherian module over itself;

2) every left (or right) ideal in $ A $ has a finite generating set;

3) every strictly ascending chain of left (or right) ideals in $ A $ breaks off after finitely many terms.

An example of a Noetherian ring is any principal ideal ring, i.e. a ring in which every ideal has one generator.

Noetherian rings are named after E. Noether, who made a systematic study of such rings and carried over to them a number of results known earlier only under more stringent restrictions (for example, Lasker's theory of primary decompositions).

A right Noetherian ring need not be left Noetherian and vice versa. For example, let $ A $ be the ring of matrices of the form

$$ \left \| \begin{array}{cc} a &\alpha \\ 0 &\beta \\ \end{array} \right \| , $$

where $ a $ is a rational integer and $ \alpha $ and $ \beta $ are rational numbers, with the usual addition and multiplication. Then $ A $ is right, but not left, Noetherian, since the left ideal of elements of the form

$$ \left \| \begin{array}{cc} 0 &\alpha \\ 0 & 0 \\ \end{array} \right \| $$

does not have a finite generating set.

Quotient rings and finite direct sums of Noetherian rings are again Noetherian, but a subring of a Noetherian ring need not be Noetherian. For example, a polynomial ring in infinitely many variables over a field is not Noetherian, although it is contained in its field of fractions, which is Noetherian.

If $ A $ is a left Noetherian ring, then so is the polynomial ring $ A [ X ] $. The corresponding property holds for the ring of formal power series over a Noetherian ring. In particular, polynomial rings of the form $ K [ X _ {1} \dots X _ {n} ] $ or $ \mathbf Z [ X _ {1} \dots X _ {n} ] $, where $ K $ is a field and $ \mathbf Z $ the ring of integers, and also quotient rings of them, are Noetherian. Every Artinian ring is Noetherian. The localization of a commutative Noetherian ring $ A $ relative to some multiplicative system $ S $ is again Noetherian. If in a commutative Noetherian ring $ A $, $ \mathfrak m $ is an ideal such that no element of the form $ 1 + m $, where $ m \in \mathfrak m $, is a divisor of zero, then $ \cap _ {k=} 1 ^ \infty \mathfrak m ^ {k} = 0 $. This means that any such ideal $ \mathfrak m $ defines on $ A $ a separable $ \mathfrak m $- adic topology. In a commutative Noetherian ring every ideal has a representation as an incontractible intersection of finitely many primary ideals. Although such a representation is not unique, the number of ideals and the set of prime ideals associated with the given primary ideals are uniquely determined.

References

[1] B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
[2] S. Lang, "Algebra" , Addison-Wesley (1974)
[3] C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973)
How to Cite This Entry:
Noetherian ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Noetherian_ring&oldid=49490
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