# Difference between revisions of "Noetherian ring"

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+ | $#C+1 = 27 : ~/encyclopedia/old_files/data/N066/N.0606850 Noetherian ring, | ||

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''left (right)'' | ''left (right)'' | ||

− | A [[Ring|ring]] | + | A [[Ring|ring]] $ A $ |

+ | satisfying one of the following equivalent conditions: | ||

− | 1) | + | 1) $ A $ |

+ | 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 $ A $ |

+ | 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 $ 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. | ||

Line 13: | Line 29: | ||

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 $ A $ |

+ | be the ring of matrices of the form | ||

− | + | $$ | |

+ | \left \| | ||

+ | \begin{array}{cc} | ||

+ | a &\alpha \\ | ||

+ | 0 &\beta \\ | ||

+ | \end{array} | ||

+ | \right \| , | ||

+ | $$ | ||

− | where | + | 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. | does not have a finite generating set. | ||

Line 25: | Line 60: | ||

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 $ 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