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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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Another proof that projective modules are flat

Let $A \xrightarrow{\phi} B$ be an injective map of $R$ modules. Let $P$ be a projective $R$ module. Let $a \in A$ and $p \in P$. And suppose there is a bi-linear map $A \otimes_R P \to Q$ sending $a$ and $p$ to something nonzero, where $Q$ is…
user062295
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Clarifications on Noether Normalization

I finished reading Noether Normalization but given that I have almost no prior algebra training I am concerned that my understanding is wrong. (Starting masters in Mathematics but previously was an engineer) The reference I took is as…
Yong Hao Ng
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Ring with the property : every power of prime ideal is primary

Is there any special name of ring (commutative ring with unity) with following property : every power of prime ideal is primary ideal? I know that every PID satisfies this property, but what else?
Seewoo Lee
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Some question on filtrations

Let $S$ be a noetherian ring and $M$ a finitely generated $S$-module. There exists a filtration by submodules $$0=M_0 \subseteq M_1\subseteq \cdots \subseteq M_r=M.$$ I want to show that for any prime ideal $P$, $\mathrm{Ann}(M) \subseteq P \iff…
Sang Cheol Lee
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Filtrations of graded modules

Let $S$ be a graded noetherian ring and $M$ a finitely generated graded $S$- module. Then I know that there exists a filtration $$0=M_0 \subseteq M_1\subseteq \cdots \subseteq M_r=M$$ by graded submodules such that for each $i$,…
Sang Cheol Lee
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What is the kernel of $R[T] \to R[w]$, $w=u/v$, $vt-u \in R[T]$ prime.

Let $R$ be an integral domain, not necessarily integrally closed. Let $0 \neq v,u \in R$. Assume that $vT-u$ is a prime element of $R[T]$ (then, by Exercise 4 page 102 in Kaplansky's book "Commutative rings", $-u,v$ is an $R$-sequence). What is the…
user237522
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Syzygies and Free Resolutions

I am going to attend a workshop on Syzygies and Free Resolutions and want to prepare for that. I haven't had introduction to the subject but I studied first course in commutative algebra. I request you to guide and suggest books for my…
Kannansoundarajan
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Intersection of height one localisations of a normal Noetherian domain

Exercise 8.3 in Kemper's A Course in Commutative Algebra: Let $R$ be a commutative normal Noetherian domain. Prove that $$R=\bigcap_{\substack{P\in\operatorname{Spec}(R)\\\operatorname{ht}(P)=1}}R_P$$ Hint: For…
user336735
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Smooth algebras

All rings are Noetherian and eft. An $A$-algebra $B$ is smooth if it is flat and the fibres are geometrically regular. I want to see some examples of this notion. So I considered the $\mathbb Z$-algebra $\mathbb Z[T]/(T^2+1)$ (Gaussian integers).…
Dieck
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If $p$ is a prime ideal then $p[X]$ is a prime ideal

If $Z$ is a ring and $p$ is a prime ideal of $Z$ then $p[X]$ is a prime ideal of $Z[X]$. Is it true or false? I believe that it is true and I try to prove it like that: Take $f(x)\in p[X]$ and suppose that $f(x)=g(x)h(x)$ then $n=\deg f=\deg…
Thế Long Lê
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Localizing a ring of invariants

Let $A$ be a domain and $G$ a finite group of automorphisms of $A$. I define $$A^G=\{a\in A\mid\sigma(a)=a ,\forall\sigma\in G\}.$$ Furthermore let $S\subset A$ be multiplicatively closed such that $\sigma(S)\subset S$ for all $\sigma\in G$ and we…
azureai
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linear functors commute with finite direct sums (in R-mod)

I'm working on Exercise 8.16 from http://web.mit.edu/18.705/www/syl11f.html. In particular: Let F : ((R-mod)) → ((R-mod)) be a linear functor. Show that F always preserves finite direct sums. ... The solution provided says: The first assertion…
unknown
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If some polynomial is in an ideal $I$, how can I write it as a linear combination of the generators of $I$?

I'm looking for a (easy) procedure of some sort. I also know a little bit of Singular and CoCoA, and was wondering if you can do that in there?
Mark
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If $R'$ is an $R$-algebra, $M,N$ are $R'$-modules, when do we have $M\otimes_{R}N\simeq M\otimes_{R'}N$ naturally?

Suppose $R$ is a commutative ring with unity, and $R'$ an $R$-algebra with structure map $\phi: R\to R'$. Let $M,N$ be two $R'$-modules. Then there exists a natural $R'$-linear (hence $R$-linear) map $$ \tau: M\otimes_{R}N\to M\otimes_{R'}N $$ the…
Hamed
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Is the localization of an injective cogenerator an injective cogenerator?

We know that in Noetherian rings any localization of an injective module is again an injective module. Is the localization of any injective cogenerator again injective cogenerator?
ramin khosravi
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