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Why noetherian rings
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While in undergraduate years, abstract stuctures are very confusing since they arise without many motivations most of the times, and find their living ground later on. I would like to understand why noetherian rings are interesting.
Principal and euclidean rings are definitely useful to have arithmetic tools and notion generalizing those of the arithmetics of integers. What about noetherian rings? It often appears in arithmetic courses, however what allows them to do that other rings do not? What are the lost properties compared to a principal ring?
In one sentence: how to grasp the meaning and interest of noetherian rings (or other structures)?
abstract-algebra number-theory ring-theory noetherian
edited 28 mins ago
Bernard
113k636104
asked 1 hour ago
TheStudent
1086
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up vote
3
down vote
favorite
While in undergraduate years, abstract stuctures are very confusing since they arise without many motivations most of the times, and find their living ground later on. I would like to understand why noetherian rings are interesting.
Principal and euclidean rings are definitely useful to have arithmetic tools and notion generalizing those of the arithmetics of integers. What about noetherian rings? It often appears in arithmetic courses, however what allows them to do that other rings do not? What are the lost properties compared to a principal ring?
In one sentence: how to grasp the meaning and interest of noetherian rings (or other structures)?
abstract-algebra number-theory ring-theory noetherian
edited 28 mins ago
Bernard
113k636104
asked 1 hour ago
TheStudent
1086
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
While in undergraduate years, abstract stuctures are very confusing since they arise without many motivations most of the times, and find their living ground later on. I would like to understand why noetherian rings are interesting.
Principal and euclidean rings are definitely useful to have arithmetic tools and notion generalizing those of the arithmetics of integers. What about noetherian rings? It often appears in arithmetic courses, however what allows them to do that other rings do not? What are the lost properties compared to a principal ring?
In one sentence: how to grasp the meaning and interest of noetherian rings (or other structures)?
abstract-algebra number-theory ring-theory noetherian
edited 28 mins ago
Bernard
113k636104
asked 1 hour ago
TheStudent
1086
While in undergraduate years, abstract stuctures are very confusing since they arise without many motivations most of the times, and find their living ground later on. I would like to understand why noetherian rings are interesting.
Principal and euclidean rings are definitely useful to have arithmetic tools and notion generalizing those of the arithmetics of integers. What about noetherian rings? It often appears in arithmetic courses, however what allows them to do that other rings do not? What are the lost properties compared to a principal ring?
In one sentence: how to grasp the meaning and interest of noetherian rings (or other structures)?
abstract-algebra number-theory ring-theory noetherian
abstract-algebra number-theory ring-theory noetherian
edited 28 mins ago
Bernard
113k636104
asked 1 hour ago
TheStudent
1086
edited 28 mins ago
Bernard
113k636104
asked 1 hour ago
TheStudent
1086
edited 28 mins ago
Bernard
113k636104
edited 28 mins ago
Bernard
113k636104
edited 28 mins ago
Bernard
113k636104
113k636104
asked 1 hour ago
TheStudent
1086
asked 1 hour ago
TheStudent
1086
asked 1 hour ago
TheStudent
1086
1086
add a comment |Â
add a comment |Â
3 Answers
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3
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Noetherian conditions bring finiteness to bear, which helps a lot.
Consider the rich theory of finite dimensional vector spaces versus the not-so-rich theory of general vector spaces, for instance.
answered 51 mins ago
lhf
158k9161376
add a comment |Â
up vote
1
down vote
Noetherian rings have a stopping condition in the form that every ascending chain of ideals becomes stationary. This is particularly interesting in computations. For instance, the polynomial ring $Bbb K[x_1,ldots,x_n]$ is noetherian which follows from the Hilbert basis theorem. This can be used to show that the Buchberger algorithm for the construction of a Gröbner basis of an ideal terminates.
answered 57 mins ago
Wuestenfux
1,326128
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1
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Noetherian rings are important because essentially every example motivating rings, namely the rings naturally considered by Number Theory and Algebraic Geometry is noetherian.
The noetherianity condition, which is usually given as the stationarity of ascending chain of ideals but in fact equivalent to the finite generation of every ideal, is a natural one encompassing all said examples and has some very important implications (to name just one, the existence of maximal ideals and thus of quotient fields).
All of this is enough motivation to make the class of noetherian rings a very important one.
answered 50 mins ago
Andrea Mori
19.1k13465
add a comment |Â
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
Noetherian conditions bring finiteness to bear, which helps a lot.
Consider the rich theory of finite dimensional vector spaces versus the not-so-rich theory of general vector spaces, for instance.
answered 51 mins ago
lhf
158k9161376
add a comment |Â
up vote
3
down vote
Noetherian conditions bring finiteness to bear, which helps a lot.
Consider the rich theory of finite dimensional vector spaces versus the not-so-rich theory of general vector spaces, for instance.
answered 51 mins ago
lhf
158k9161376
add a comment |Â
up vote
3
down vote
up vote
3
down vote
Noetherian conditions bring finiteness to bear, which helps a lot.
Consider the rich theory of finite dimensional vector spaces versus the not-so-rich theory of general vector spaces, for instance.
answered 51 mins ago
lhf
158k9161376
Noetherian conditions bring finiteness to bear, which helps a lot.
Consider the rich theory of finite dimensional vector spaces versus the not-so-rich theory of general vector spaces, for instance.
answered 51 mins ago
lhf
158k9161376
answered 51 mins ago
lhf
158k9161376
answered 51 mins ago
lhf
158k9161376
answered 51 mins ago
lhf
158k9161376
158k9161376
add a comment |Â
add a comment |Â
up vote
1
down vote
Noetherian rings have a stopping condition in the form that every ascending chain of ideals becomes stationary. This is particularly interesting in computations. For instance, the polynomial ring $Bbb K[x_1,ldots,x_n]$ is noetherian which follows from the Hilbert basis theorem. This can be used to show that the Buchberger algorithm for the construction of a Gröbner basis of an ideal terminates.
answered 57 mins ago
Wuestenfux
1,326128
add a comment |Â
up vote
1
down vote
Noetherian rings have a stopping condition in the form that every ascending chain of ideals becomes stationary. This is particularly interesting in computations. For instance, the polynomial ring $Bbb K[x_1,ldots,x_n]$ is noetherian which follows from the Hilbert basis theorem. This can be used to show that the Buchberger algorithm for the construction of a Gröbner basis of an ideal terminates.
answered 57 mins ago
Wuestenfux
1,326128
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Noetherian rings have a stopping condition in the form that every ascending chain of ideals becomes stationary. This is particularly interesting in computations. For instance, the polynomial ring $Bbb K[x_1,ldots,x_n]$ is noetherian which follows from the Hilbert basis theorem. This can be used to show that the Buchberger algorithm for the construction of a Gröbner basis of an ideal terminates.
answered 57 mins ago
Wuestenfux
1,326128
Noetherian rings have a stopping condition in the form that every ascending chain of ideals becomes stationary. This is particularly interesting in computations. For instance, the polynomial ring $Bbb K[x_1,ldots,x_n]$ is noetherian which follows from the Hilbert basis theorem. This can be used to show that the Buchberger algorithm for the construction of a Gröbner basis of an ideal terminates.
answered 57 mins ago
Wuestenfux
1,326128
answered 57 mins ago
Wuestenfux
1,326128
answered 57 mins ago
Wuestenfux
1,326128
answered 57 mins ago
Wuestenfux
1,326128
1,326128
add a comment |Â
add a comment |Â
up vote
1
down vote
Noetherian rings are important because essentially every example motivating rings, namely the rings naturally considered by Number Theory and Algebraic Geometry is noetherian.
The noetherianity condition, which is usually given as the stationarity of ascending chain of ideals but in fact equivalent to the finite generation of every ideal, is a natural one encompassing all said examples and has some very important implications (to name just one, the existence of maximal ideals and thus of quotient fields).
All of this is enough motivation to make the class of noetherian rings a very important one.
answered 50 mins ago
Andrea Mori
19.1k13465
add a comment |Â
up vote
1
down vote
Noetherian rings are important because essentially every example motivating rings, namely the rings naturally considered by Number Theory and Algebraic Geometry is noetherian.
The noetherianity condition, which is usually given as the stationarity of ascending chain of ideals but in fact equivalent to the finite generation of every ideal, is a natural one encompassing all said examples and has some very important implications (to name just one, the existence of maximal ideals and thus of quotient fields).
All of this is enough motivation to make the class of noetherian rings a very important one.
answered 50 mins ago
Andrea Mori
19.1k13465
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Noetherian rings are important because essentially every example motivating rings, namely the rings naturally considered by Number Theory and Algebraic Geometry is noetherian.
The noetherianity condition, which is usually given as the stationarity of ascending chain of ideals but in fact equivalent to the finite generation of every ideal, is a natural one encompassing all said examples and has some very important implications (to name just one, the existence of maximal ideals and thus of quotient fields).
All of this is enough motivation to make the class of noetherian rings a very important one.
answered 50 mins ago
Andrea Mori
19.1k13465
Noetherian rings are important because essentially every example motivating rings, namely the rings naturally considered by Number Theory and Algebraic Geometry is noetherian.
The noetherianity condition, which is usually given as the stationarity of ascending chain of ideals but in fact equivalent to the finite generation of every ideal, is a natural one encompassing all said examples and has some very important implications (to name just one, the existence of maximal ideals and thus of quotient fields).
All of this is enough motivation to make the class of noetherian rings a very important one.
answered 50 mins ago
Andrea Mori
19.1k13465
edited 39 mins ago
answered 50 mins ago
Andrea Mori
19.1k13465
answered 50 mins ago
Andrea Mori
19.1k13465
answered 50 mins ago
Andrea Mori
19.1k13465
19.1k13465
add a comment |Â
add a comment |Â
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