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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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Noetherian Endomorphisms

Suppose $R$ is Noetherian and $M$ is a finitely generated $R$-module. IF $g \in \text {End}_R(M,M)$ does there exist a k such that $g^k =g^{k+1}= \ldots$ I'm trying to work with the generators, but not getting anywhere. Any assistance would be…
user92612
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Is a special module homomorphism injective?

Let $I$ be an ideal in a ring $B$ with $I^2=0$. Furthermore one knows that one has a splitting $\alpha: B/I \rightarrow B$ of the natural projection. Let $M$ be a finitely generated module over $B/I$. Set $N:= M \otimes_{B/I} I$, where we let $I$ be…
Void
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proving that the height of $(X)$ in $R[X]$ is equal to 1

Let $R$ be a Noetherian ring. I want to prove that $\operatorname{height}(X) = 1$. Here is how i do it: It is not hard to show that $\operatorname{height}(X) = \operatorname{height}(p,X)$, where $p$ is a minimal prime of $R$. Since the…
Manos
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a "strange" depth inequality

The following question arises in the context of the proof of Proposition 3.3.18 in Bruns and Herzog, Cohen-Macaulay Rings. Let $R$ be a CM ring, not necessarily local, and suppose that $R$ admits a canonical module $\omega_R$, i.e. for every maximal…
Manos
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Reference for $(N_1\cap N_2)\otimes_A M = (N_1\otimes_A M)\cap ( N_2\otimes_A M)$

Where can I find a canonical proof of the following statement? If $M$ is a flat $A$-module and $N$ is an $A$-module with submodules $N_1, N_2$, then $$(N_1\cap N_2)\otimes_A M = (N_1\otimes_A M)\cap (N_2\otimes_A M)$$ inside $N \otimes_A M$. This…
going flatinum
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existence of a finite-length maximal regular sequence

Theorem 16.7 in Matsumura's Commutative Ring Theory reads as follows: "Let $A$ be a Noetherian ring, $I$ an ideal of $A$ and $M$ a finite $A$-module such that $IM \neq M$; then the length of a maximal $M$-sequence in $I$ is a well determined integer…
Manos
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completion of the canonical module

For a local Noetherian Cohen-Macaulay ring $(R,m,k)$ the canonical module is defined to be any maximal Cohen-Macaulay module of finite injective dimension and of type $1$. The canonical module is unique up to isomorphism. So let $\omega_R$ be the…
Manos
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$(x_1,\ldots x_n)=(1)\implies (x_1^{k_1},\ldots, x^{k_n})=(1)$

Let $R$ be a commutative ring with unit, I'm trying to prove why in this ring $$(x_1,\ldots x_n)=(1)\implies (x_1^{k_1},\ldots, x^{k_n})=(1)$$ It seems an easy question, but I couldn't prove it, I need a hint or something. Thanks
user108205
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Associated prime preserved under the quotient

Let $(R,m,k)$ be a complete local Noetherian ring and let $E$ be an $R$-module such that $\operatorname{Ass}E=\left\{m\right\}$. Let $N$ be a proper submodule of $E$. Question: Is it true that $\operatorname{Ass}E/N=\left\{m\right\}$ and why? If it…
Manos
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combinatorial commutative algebra

Is there anyone who can help me with this problem? Any hint to the solution would be appreciated! Let $\Delta$ be a $(d-1)$-dimensional simplicial complex. Show that the h- and f-vectors of $\Delta$ satisfy the following…
M.D.D
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Can one find in this specific setting an extension of a given ring map?

All rings in this question are unitary and commutative and all maps are homomorphisms of commutative rings sending $1$ to $1$. Let $R$ and $S$ be regular local rings and let $$ \begin{array}{rcl} && R[x,y,z]/(x+y+z-1)\\ &&\qquad\qquad…
sopot
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Homomorphism between finitely generated $k$-algebras of (Krull) dimension 1 preserving maximal properties of ideals.

Let $k$ be any field (may not be algebraically closed), $A$ and $B$ are two finitely-generated $k$-algebras of (Krull) dimension 1. Suppose $f : A \rightarrow B $ be a $k$-algebra homomorphism. I want to show that if $M$ is any maximal ideal in $B$,…
Peter Hu
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Induced map on spectrum of the rings on integral extensions

I know that if $A$ is contained in $B$ and $B$ is an integral extension of $A$, then the induced map on spectrum of the rings is surjective (and closed). Is it true if $B$ is not assumed to be integral over $A$?
MathStudent
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local ring has same dimension as its regular local subring

Let $R$ be a Noetherian local ring and $S$ a regular local subring of $R$ such that $R$ is a finite $S$-module. Question: Why is it true that every regular system of parameters of $S$ is a system of parameters of $R$? Motivation: Proof of Prp.…
Manos
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faithful, finitely generated module over a local ring

Let $A$ be a commutative local ring, with unique maximal ideal $\mathfrak{m}$, and residue field $k:=A/\mathfrak{m}$. Let $M$ be a faithful, finitely generated $A$-module. If $M/\mathfrak{m}M$ is 2-dimensional over $k$, is $M$ necessarily free over…
Dave
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