Does $DeclareMathOperator{len}{length}DeclareMathOperator{rk}{rank}len(M/xM) leq rk(M) cdot len(R/(x))$ hold...
$begingroup$
$DeclareMathOperator{len}{length}
DeclareMathOperator{rk}{rank}$In Eisenbud's book Commutative Algebra with a View towards Algebraic Geometry he says:
The basic result of this section expresses the length of $M/aM$ for certain $R$-modules $M$ in terms of the length of $R/(a)$ and an invariant of $M$. [...] For simplicity, and because it suffices for the application, we shall assume here that $R$ is a domain.
He then defines the rank of $M$ over $R$ as $rk(M) = dim_{K} K otimes_R M$ where $K$ is the field of fractions of $R$. After that he goes on and gives the following result:
Lemma 11.12. Let $R$ be a one-dimensional Noetherian domain. If $M$ is a torsion-free $R$-module, then $$len(M/xM) leq rk(M) cdot len(R/(x))$$ with equality if $M$ is finitely generated as an $R$-module.
Now I am wondering if there holds a similar result in the case when $R$ has zero divisors. But then, if I recall correctly, the rank is only locally well-defined.
So, can anyone give me references where this is handled in the non-integral case? Thank you very much in advance!
ring-theory commutative-algebra modules
asked Dec 10 '18 at 12:59
windsheafwindsheaf
612312
$endgroup$
add a comment |
$begingroup$
$DeclareMathOperator{len}{length}
DeclareMathOperator{rk}{rank}$In Eisenbud's book Commutative Algebra with a View towards Algebraic Geometry he says:
The basic result of this section expresses the length of $M/aM$ for certain $R$-modules $M$ in terms of the length of $R/(a)$ and an invariant of $M$. [...] For simplicity, and because it suffices for the application, we shall assume here that $R$ is a domain.
He then defines the rank of $M$ over $R$ as $rk(M) = dim_{K} K otimes_R M$ where $K$ is the field of fractions of $R$. After that he goes on and gives the following result:
Lemma 11.12. Let $R$ be a one-dimensional Noetherian domain. If $M$ is a torsion-free $R$-module, then $$len(M/xM) leq rk(M) cdot len(R/(x))$$ with equality if $M$ is finitely generated as an $R$-module.
Now I am wondering if there holds a similar result in the case when $R$ has zero divisors. But then, if I recall correctly, the rank is only locally well-defined.
So, can anyone give me references where this is handled in the non-integral case? Thank you very much in advance!
ring-theory commutative-algebra modules
asked Dec 10 '18 at 12:59
windsheafwindsheaf
612312
$endgroup$
$begingroup$
How would you define rank locally when $R$ has zero divisors? Can you give an example?
$endgroup$
– Youngsu
Dec 10 '18 at 23:24
$begingroup$
@Youngsu I don't know. Do you know any references where something familiar to the rank is defined for modules over non-integral rings?
$endgroup$
– windsheaf
Dec 11 '18 at 8:13
$begingroup$
@Youngsu I guess I was thinking about the possibility (when I said I've heard that the rank is locally well-defined) to define the rank locally be the minimal number of generators.
$endgroup$
– windsheaf
Dec 11 '18 at 8:23
add a comment |
$begingroup$
$DeclareMathOperator{len}{length}
DeclareMathOperator{rk}{rank}$In Eisenbud's book Commutative Algebra with a View towards Algebraic Geometry he says:
The basic result of this section expresses the length of $M/aM$ for certain $R$-modules $M$ in terms of the length of $R/(a)$ and an invariant of $M$. [...] For simplicity, and because it suffices for the application, we shall assume here that $R$ is a domain.
He then defines the rank of $M$ over $R$ as $rk(M) = dim_{K} K otimes_R M$ where $K$ is the field of fractions of $R$. After that he goes on and gives the following result:
Lemma 11.12. Let $R$ be a one-dimensional Noetherian domain. If $M$ is a torsion-free $R$-module, then $$len(M/xM) leq rk(M) cdot len(R/(x))$$ with equality if $M$ is finitely generated as an $R$-module.
Now I am wondering if there holds a similar result in the case when $R$ has zero divisors. But then, if I recall correctly, the rank is only locally well-defined.
So, can anyone give me references where this is handled in the non-integral case? Thank you very much in advance!
ring-theory commutative-algebra modules
asked Dec 10 '18 at 12:59
windsheafwindsheaf
612312
$endgroup$
$DeclareMathOperator{len}{length}
DeclareMathOperator{rk}{rank}$In Eisenbud's book Commutative Algebra with a View towards Algebraic Geometry he says:
The basic result of this section expresses the length of $M/aM$ for certain $R$-modules $M$ in terms of the length of $R/(a)$ and an invariant of $M$. [...] For simplicity, and because it suffices for the application, we shall assume here that $R$ is a domain.
He then defines the rank of $M$ over $R$ as $rk(M) = dim_{K} K otimes_R M$ where $K$ is the field of fractions of $R$. After that he goes on and gives the following result:
Lemma 11.12. Let $R$ be a one-dimensional Noetherian domain. If $M$ is a torsion-free $R$-module, then $$len(M/xM) leq rk(M) cdot len(R/(x))$$ with equality if $M$ is finitely generated as an $R$-module.
Now I am wondering if there holds a similar result in the case when $R$ has zero divisors. But then, if I recall correctly, the rank is only locally well-defined.
So, can anyone give me references where this is handled in the non-integral case? Thank you very much in advance!
ring-theory commutative-algebra modules
ring-theory commutative-algebra modules
asked Dec 10 '18 at 12:59
windsheafwindsheaf
612312
asked Dec 10 '18 at 12:59
windsheafwindsheaf
612312
asked Dec 10 '18 at 12:59
windsheafwindsheaf
612312
asked Dec 10 '18 at 12:59
windsheafwindsheaf
612312
asked Dec 10 '18 at 12:59
windsheafwindsheaf
612312
612312
$begingroup$
How would you define rank locally when $R$ has zero divisors? Can you give an example?
$endgroup$
– Youngsu
Dec 10 '18 at 23:24
$begingroup$
@Youngsu I don't know. Do you know any references where something familiar to the rank is defined for modules over non-integral rings?
$endgroup$
– windsheaf
Dec 11 '18 at 8:13
$begingroup$
@Youngsu I guess I was thinking about the possibility (when I said I've heard that the rank is locally well-defined) to define the rank locally be the minimal number of generators.
$endgroup$
– windsheaf
Dec 11 '18 at 8:23
add a comment |
$begingroup$
How would you define rank locally when $R$ has zero divisors? Can you give an example?
$endgroup$
– Youngsu
Dec 10 '18 at 23:24
$begingroup$
@Youngsu I don't know. Do you know any references where something familiar to the rank is defined for modules over non-integral rings?
$endgroup$
– windsheaf
Dec 11 '18 at 8:13
$begingroup$
@Youngsu I guess I was thinking about the possibility (when I said I've heard that the rank is locally well-defined) to define the rank locally be the minimal number of generators.
$endgroup$
– windsheaf
Dec 11 '18 at 8:23
$begingroup$
How would you define rank locally when $R$ has zero divisors? Can you give an example?
$endgroup$
– Youngsu
Dec 10 '18 at 23:24
$begingroup$
How would you define rank locally when $R$ has zero divisors? Can you give an example?
$endgroup$
– Youngsu
Dec 10 '18 at 23:24
$begingroup$
@Youngsu I don't know. Do you know any references where something familiar to the rank is defined for modules over non-integral rings?
$endgroup$
– windsheaf
Dec 11 '18 at 8:13
$begingroup$
@Youngsu I don't know. Do you know any references where something familiar to the rank is defined for modules over non-integral rings?
$endgroup$
– windsheaf
Dec 11 '18 at 8:13
$begingroup$
@Youngsu I guess I was thinking about the possibility (when I said I've heard that the rank is locally well-defined) to define the rank locally be the minimal number of generators.
$endgroup$
– windsheaf
Dec 11 '18 at 8:23
$begingroup$
@Youngsu I guess I was thinking about the possibility (when I said I've heard that the rank is locally well-defined) to define the rank locally be the minimal number of generators.
$endgroup$
– windsheaf
Dec 11 '18 at 8:23
add a comment |
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$begingroup$
How would you define rank locally when $R$ has zero divisors? Can you give an example?
$endgroup$
– Youngsu
Dec 10 '18 at 23:24
$begingroup$
@Youngsu I don't know. Do you know any references where something familiar to the rank is defined for modules over non-integral rings?
$endgroup$
– windsheaf
Dec 11 '18 at 8:13
$begingroup$
@Youngsu I guess I was thinking about the possibility (when I said I've heard that the rank is locally well-defined) to define the rank locally be the minimal number of generators.
$endgroup$
– windsheaf
Dec 11 '18 at 8:23