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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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Fraction ring contains another implies prime contains another

As part of lemma 6.4 in Hartshorne, I came across a statement that I can't prove Let $m,n $ be maximal ideals of an integral domain $A$. Then $ A_m \subset A_n$ implies $n \subset m $. It is clear intuitively. To prove it, I picked $s \in A-m$.…
PeterM
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Contraction of non-zero prime ideals in the ring of algebraic integers

Let $A= \{y \in \mathbb{C} :$ $y$ integral over $\mathbb{Z}$ }. Let $P\not=\{0 \}$ be a prime ideal of $A$. Prove that $P \cap \mathbb{Z} \not= \{0 \}$. Iam totally stuck here, it is given that $P$ contains an element $a \not= 0$, I tried to assume…
user117449
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If $M$ is a flat $R$-module, is $M/IM$ a flat $R/I$-module?

Let $R$ be a Noetherian (local) ring, and let $M$ be a finitely generated, flat $R$-module. Further, let $I$ be an ideal of $R$. Question: Is $M/IM$ flat over $R/I$?
Sebastian
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every ideal that is not intersect $S$ is prime ideal?

every ideal that is not intersect $S$ where $S$ is multiplicative closed is prime ideal? I know that maximal ideals among those are prime ideals. But what about other ideals that is not intersect $S$. Are they also prime ideal?
claire
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Length of polynomial ring modulo a homogeneous regular sequence

Proposition: Let $k$ be a field and $R=k[x_1,\dots,x_n]$ the polynomial ring with $x_i$ having degree $1$. Let $f_1,\dots,f_n$ be homogeneous elements such that $\deg(f_i)=s_i >0$ and they form an $R$-sequence. Then the length of…
Manos
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Maximal $R$-sequences in ideals

If $\alpha_1,...,\alpha_s$ is a maximal $R$-sequence in an ideal $I$ ($R$ is commutative with unity), is this always true that $I⊆P$, where $P\in\operatorname{Ass} (\alpha_1,...,\alpha_s)$? In case $R$ is Noetherian this is true because, by prime…
karparvar
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A vector space in the form of a tensor product

Let $R$ be a commutative domain with fraction field $K$. It is known that $K_R$ is injective. Now, if $M_R$ is a torsion-free module and we localize at $S=R-0$ we get $M⊗_RK=S^{-1}M⊇M$. My question is: Why $M⊗_RK$ is a $K$- vector space, and what is…
karparvar
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Show that $A(V(f))/(X,Y)^n A(V(f))$is a local ring where $A(V(f))$ is the coordinate ring of an irreducible polynomial $f$ in $K[X,Y]$

Let $K$ be an algebraic closed field, $f$ is an irreducible polynomial in $K[X,Y]$,and $f(0,0)=0$. Denote $A(V(f))$ as the coordinate ring $K[X,Y]/(f)$. Now I don't konw how to show that $A(V(f))/(X,Y)^n A(V(f))$ is a local ring. Can any one give me…
molan
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Questions regarding a proof of Nakayama's lemma.

I refer to this proof of Nakayama's lemma. What is $\varphi^n$? Is it $\underbrace{\varphi\circ\varphi\circ\dots\circ\varphi}_{\text{$n$ times}}$? What is $\varphi\delta_{ij}$?
user67803
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A question about one of Hartshorne's propositions

Hartshorne says that for $S_1,S_2\in A[x_1,x_2,\dots,x_n]$, where $A$ is a commutative ring, $Z(S_1)\cup Z(S_2)=Z(S_1S_2)$. Shouldn't it be $Z(S_1)\cup Z(S_2)=Z(S_1\cap S_2)$? We know that $S_1S_2\subseteq S_1\cap S_2$.
freebird
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$\mathbb{Q}[x,1/x]$ is normal?

Let $x$ be a transcendental. I heard $\mathbb{Q}[x,1/x]$ is a normal domain. But I don't understand why. Help me, thanks.
user55871
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A quotient of a regular local ring may not be regular

Let $(R,m)$ be a regular local ring having an ideal $I$ such that $I$ is a subset of $m^2$. If $I$ possesses a non-zerodivisor, I want to show that $R/I$ can not be regular. My try is just that $m$ could be generated by an $R$- sequence with…
karparvar
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Grade of maximal ideals in polynomial rings over Artinian local rings

If $R$ is a commutative Artinian ring it is well-known that $R$ is Cohen-Macaulay. Also, if $S$ is a Cohen-Macaulay ring, then any polynomial ring $S[X_1,\dots,X_n]$ is so. Now if $R$ is a commutative Artinian local ring, how we could derive that…
karparvar
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Noetherian Jacobson rings

One of the equivalent forms in definition of a Noetherian Jacobson ring $R$ is that $R$ has no prime ideals $P$ such that $R/P$ is a 1-dimensional semi-local ring. When $R/P$ has dimension 1, it means that between $R$ and $P$ there is at most one…
karparvar
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How to understand the regular sequence of a module

If we have a regular sequence $a_1,\dots, a_r$ in a ring $A$, I think it means the subschemes $A/(a_1,\dots,a_i)$ cut out step by step are all equi-dimensional. (when $A$ is affine coordinate ring, by the krull principal ideal theorem). But what…
user93417
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