Difference between revisions of "Noetherian ring"
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''left (right)'' | ''left (right)'' | ||
− | A [[Ring|ring]] | + | 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: |
− | satisfying one of the following equivalent conditions: | ||
− | 1) | + | 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; |
− | is a left (or right) [[Noetherian module|Noetherian module]] over itself; | ||
− | 2) every left (or right) ideal in | + | 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; |
− | has a finite generating set; | ||
− | 3) every strictly ascending chain of left (or right) ideals in | + | 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. |
− | 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 | + | 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 |
− | 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> | |
− | |||
− | where | + | 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 |
− | is a rational integer and | ||
− | and | ||
− | are rational numbers, with the usual addition and multiplication. Then | ||
− | 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> | |
− | |||
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 | + | 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. |
− | is a left Noetherian ring, then so is the polynomial ring | ||
− | The corresponding property holds for the ring of formal power series over a Noetherian ring. In particular, polynomial rings of the form | ||
− | or | ||
− | where | ||
− | is a field and | ||
− | 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 | ||
− | relative to some multiplicative system | ||
− | is again Noetherian. If in a commutative Noetherian ring | ||
− | |||
− | is an ideal such that no element of the form | ||
− | where | ||
− | is a divisor of zero, then | ||
− | This means that any such ideal | ||
− | defines on | ||
− | a separable | ||
− | 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> |
Revision as of 14:52, 7 June 2020
left (right)
A ring satisfying one of the following equivalent conditions:
1) is a left (or right) Noetherian module over itself;
2) every left (or right) ideal in has a finite generating set;
3) every strictly ascending chain of left (or right) ideals in 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 be the ring of matrices of the form
![]() |
where is a rational integer and
and
are rational numbers, with the usual addition and multiplication. Then
is right, but not left, Noetherian, since the left ideal of elements of the form
![]() |
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 is a left Noetherian ring, then so is the polynomial ring
. The corresponding property holds for the ring of formal power series over a Noetherian ring. In particular, polynomial rings of the form
or
, where
is a field and
the ring of integers, and also quotient rings of them, are Noetherian. Every Artinian ring is Noetherian. The localization of a commutative Noetherian ring
relative to some multiplicative system
is again Noetherian. If in a commutative Noetherian ring
,
is an ideal such that no element of the form
, where
, is a divisor of zero, then
. This means that any such ideal
defines on
a separable
-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) |
Noetherian ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Noetherian_ring&oldid=47979