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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.

16857 questions
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Exactness of Dual Sequence, A Proposition in Atiyah and MacDonald

The proposition 2.9 of Atiyah and Macdonald syas that a sequence of $A$-modules $$M'\xrightarrow u M \xrightarrow v M'' \rightarrow 0$$ is exact iff the dual sequence $$0\rightarrow Hom (M'',N)\xrightarrow{\bar{v}} Hom(M,N)\xrightarrow{\bar{u}} Hom…
Michelle
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Another question about proposition 5.15 of Atiyah-MacDonald.

I have a trouble in understanding the proposition 5.15 in the book from Atiyah and MacDonald. I see that some time ago another user asked a similar question (Proposition 5.15 Atiyah Macdonald: Integral Closure and Minimal Polynomial) but…
user233650
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Localization of an integral A-algebra is not always integral.

Let $A$, $B$ rings with a morphism $f : A \to B$ and suppose that $B$ is integral over $A$. Let $\mathfrak{n} \subseteq B$ a maximal ideal, and $\mathfrak{m}$ its preimage under $f$ (so $\mathfrak{m}$ is maximal in $A$). The question is: Is the…
user233650
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The Dimension Sequence of a Ring

Let $R$ be a commutative ring of finite Krull dimension $n_0$. Let $\dim(R[X_1,\dots,X_m])=n_m$. The sequence $\{n_i\}_{i=0}^\infty$ is called the dimension sequence of $R$. Let $d_i=n_i-n_{i-1}$. The sequence $\{d_i\}_{i=1}^\infty$ is called the…
Andrew Chiriac
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A question on additive-functions in the proof of the Hilbert-Serre Theorem

I am trying to understand the proof of the following Theorem from Atiyah-MacDonald. $P(M,t)$ is a rational function in t of the form $f(t)/\prod_{i=1}^{s}(1-t^{k_{i}})$ $P(M,t)=\sum_{n=0}\lambda(M_{n})t^{n}$ is the Poincare Series and $\lambda$ is…
user135520
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No ring isomorphism between certain rings

Let $k$ be an algebraically closed field and let$c,d$ be distinct elements of $k$. Why there is no ring isomorphism between $k[x,\frac{1}{x}]$ and $k[x,(x-c)^{-1},(x-d)^{-1}]$? I guess one approach is to look at the units, however (just for…
user6495
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Contraction of a maximal ideal in a polynomial ring

I have two questions: If $K$ is a field, $R=K[x_1,\ldots,x_n]$, the ring of polynomials over $K$ with $n$ indeterminates, and $M$ is a maximal ideal of $R$ why is the contraction $N$ of $M$ to $K[x_1,\ldots,x_{n-1}]$ maximal? Why $M$ lies properly…
karparvar
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on generators of $k$-algebra

Let $(A,m,k)$ be a local ring, and $A$ is a finitely generated $k$-algebra, and the maximal ideal $m$ is nilpotent. Let $x_1,\ldots,x_n \in m$ and their canonical images in $m/m^2$ generate this $k$-vector space. How to show that $x_1,\ldots,x_n$…
John
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A normality criterion

I'm trying to solve exercise 8.5 from the book "A course in Commutative algebra" by Gregor Kemper. It says Let $R$ be a ring and $a\in R$ such that $a$ is not a zero divisor, the ideal $(a)$ is a radical ideal and the localization $R_a$…
Nikitas Nikandros
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$A\subset B $ with $B$ integral domain. If $B$ is integral over $A$ can we say that $Q(B)$ is algebraic over $Q(A)$?

Let $A\subset B$ with $B$ an integral domain. If $B$ is integral over $A$ can we say that $Q(B)$ is algebraic over $Q(A)$ ? (Here $Q(\dots)$ denotes the quotient field of $(\dots))$.)
Via
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Homological criterion for $A(B \cap C) = AB \cap AC$?

Is there a homological criterion for the condition $A(B \cap C) = AB \cap AC$ for ideals in a ring $R$? By "homological" I mean a statement such as "the given equation holds if and only if (some Tor, Ext, local cohomology, etc) group vanishes/does…
Jake Levinson
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Finding the kernel of a multiplication map

Consider the ideal $I=(x,y) \subset R=\mathbb{C}[x,y]$ and $\mathbb{C}$ as the $R$-module $R/I$. I am asked to find the kernel of the multiplication map $I \otimes_R I \rightarrow I$ as a submodule of $I$. I know this is the ideal generated by…
Leo163
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Example of filtration which is not stable

The following proposition was given in Liu's Algebraic Geometry and Arithmetic Curves: Let $A$ be a Noetherian ring, $I$ an ideal of $A$, and $M$ a finitely generated $A$-module endowed with stable $I$-filtration $(M_n)_n$. Then for any submodule…
Jaakko Seppälä
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Ring of germs and induced isomorphism

Let $A$ be the ring of germs of real analytic functions in $0\in\mathbb R$. Let $x\in A$ be the identity map on $\mathbb R$. How can I show that the map $f\mapsto \sum_n(f^{(n)}(0)/n!)T^n$ from $A$ to $\mathbb R[[T]]$ is injective, and induces an…
Jaakko Seppälä
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Is $\operatorname{Hom}_R(R/m,R/(x_1,...,x_d))$ isomorphic to $R/m$?

Let $(R,m)$ be a local ring. Let $x_1,...,x_d$ be a maximal $R$-sequence. Is $\operatorname{Hom}_R(R/m,R/(x_1,...,x_d))$ isomorphic to $R/m$?
user177523
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