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Questions tagged [commutative-algebra]

Questions about commutative rings, their ideals, and their modules.

Commutative algebra is the area of mathematics that deals with commutative rings and their ideals, as well as modules over commutative rings.

Many results and tools of commutative algebra are cornerstones of algebraic geometry. Important tools of commutative algebra include localization and completion of rings and modules.

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A characterization of invertible fractional ideals of an integral domain

Let $A$ be an integral domain, $K$ its field of fractions. Let $M$ be a fractional ideal of $A$. I'd like to prove that $M$ is invertible if and only if $MA_P$ is a principal fractional ideal of $A_P$ for every maximal ideal $P$ of $A$. EDIT As…
Makoto Kato
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Understanding the conductor ideal of a ring.

Consider the inclusion of a ring $A$ into its integral closure $B$. The conductor ideal $I$ is defined as $I:=\{a\in A~|~aB\subseteq A\}$. This is supposed to describe the locus where the normalization map $\textrm{Spec}(B)\rightarrow…
jrajchgot
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Prime ideals and irreducible ideals

I have the following definitions: An ideal $I$ is prime if, whenever $ab \in I$, either $a \in I$ or $b \in I$. An ideal $J$ is irreducible if, whenever $J = I_1 \cap I_2$ for ideals $I_1 $ and $ I_2$, $J \subseteq I_1$ or $J \subseteq I_2$. EDIT:…
Matt
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Finite length modules over local rings

Let $A$ be a noetherian local ring and $M$ be an artinian and noetherian module over $A$. Does one know a priori anything about the structure of $M$? Furthermore: if one knows that the length of $M$ as $A$-module is $1$, i.e. $M$ is simple over $A$,…
Cyril
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The Ring of Cauchy Sequences

Let $S$ be the ring of Cauchy sequences of $\mathbb{Q}$, i.e. $S=\{(a_n)\in\mathbb{Q}^{\mathbb{N}}|(a_n)\, \text{is a Cauchy rational sequence in the ordinary distance} \}$, $S$ is a subring of $\mathbb{Q}^{\mathbb{N}}$. Denote $R$ to the ring…
wxu
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Converse of fundamental theorem of finitely-generated modules

Let $R$ be a ring. If every finitely-generated $R$-module $M$ is isomorphic to a finite direct product of quotients of $R$ by ideals then call $R$ a wheel ring. For a domain $R$ we have the implications $$\rm PID\implies wheel\implies Bezout $$ The…
anon
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Modules over $k[X,Y]$

Over a PID like $k[X]$, all (non-trivial) ideals are free and hence projective. But the ring $k[X,Y]$ is not a PID. Is it possible to describe all ideals of this particular ring which are projective modules? (What if we restrict to $k$ algebraically…
Evariste
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Ideal generated by a irreducible element

Is the ideal generated by an irreducible element always a prime ideal in a ring? If so why?
Mohan
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Proof that a certain derivation is well defined

I have spent several hours on this, apparently straightforward issue. This is with reference to page 17 in the following notes http://www.math.lsa.umich.edu/~hochster/615W10/615.pdf Suppose, $R$ is a commutative ring, $W$ a multiplicatively closed…
MVK
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Units in formal power series and formal Laurent series rings

Let $R$ be a commutative ring with unit, $R[[t]]$ the ring of formal power series over $R$ and $R((t))$ the ring of formal Laurent series of $R$. It is easy to see (and well known) that the group of units $R[[t]]^{\times}$ equals…
KotelKanim
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A mistake in the proof of Lazard's theorem in Eisenbud's book on Commutative Algebra?

In Prof. Eisenbud's book Commutative Algebra with a View Toward Algebraic Geometry, there is in Appendix 6.2 a proof of Govorov & Lazard theorem that seems to me slightly wrong. It is written (last line of page 712 and first line of page 713 of my…
brunoh
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Finite ring extension and number of maximal ideals

I want to understand why the following is true: Let $S \subseteq R$ be commutative rings with $1$ and assume that $R$ is finitely generated as an $S$-module by at most $k$ elements. For every maximal ideal $M$ of $S$ there are at most $k$ maximal…
user6495
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If a maximal ideal $\mathfrak m$ is flat, then $\dim_k (\mathfrak m/\mathfrak m^2) \leq 1$

This is an exercise that bothers me a lot: Let $R$ be a commutative ring with $1$. Let $\mathfrak{m}$ be a maximal ideal in $R$. If $\mathfrak m$ is flat as an $R$-module then the vector space dimension $\dim_k(\mathfrak{m}/\mathfrak{m}^2) \leq…
Louis
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Counterexample for going up theorem

I am searching for an example which shows that integral extensions are necessary for going up theorem. Basically I want rings $A\subset B$ (not integral extension) such that lying over holds, but going up does not hold.
hjmm
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Intersection of two localizations

Let $A$ be a commutative ring with unity. If $\mathfrak p,\mathfrak q\in \operatorname{Spec} (A)$ is it true the following equality $$A_\mathfrak p\cap A_\mathfrak q= A_{\mathfrak p\cup \mathfrak q }?$$ Here the symbol $A_{\mathfrak p\cup…
Dubious
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