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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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Finite extension of integrally closed ring again integrally closed

Let $S\subset R$ be a finite ring extension, i.e. $R$ is finitely generated as an $S$-module. Assume that $S$ is integrally closed. Does this imply that also $R$ is integrally closed (in its quotient field)?
Sebastian
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a case where contraction of a principal ideal is principal

Let $K$ be a field and $R_1,\cdots,R_n$ DVRs of $K$ with $m_i$ the maximal ideal of $R_i$. Define $A=\cap R_i$. Then $A$ is semilocal with maximal ideals $p_i=m_i \cap A$. Also, $A_{p_i} = R_i$. Question: How can we see that $p_i$ is principal? My…
Manos
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$0\to L\to R^{n}\to M \to 0$ is exact, prove $M$ is finitely presented if and only if $L$ is finitely generated.

Suppose $R$ is a ring, $0 \rightarrow L\rightarrow R^{n} \rightarrow M \rightarrow 0$ is a short exact sequence, prove $M$ is finitely presented if and only if $L$ is finitely generated.
Alex
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fiber product of local artinian rings

Let $A,B,C$ be local artinian rings and $p : A \to C, q : B \to C$ local homomorphisms. Why is the fiber product $A \times_C B$ again a local artinian ring? It is easy to see that $P:=A \times_C B$ is a local ring with maximal ideal…
Martin Brandenburg
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Associated Primes of Tensor Product

Let $R$ be a Noetherian ring, and let $M$ and $N$ be finitely generated $R$ module. Do we know any formulas for $\operatorname{Ass}(M\otimes_R N)$ in terms of $\operatorname{Ass}(M)$, $\operatorname{Ass}(N)$ or in terms of $\operatorname{Supp}(M)$…
messi
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Verifying Hilberts Nullstellensatz on a particular example

Let $k$ be an algebraically closed field of characteristic $2$ and consider the following equations: $$xy + z^2 = 0$$ $$uv + w^2 = 0$$ $$uy + vx = 0$$ It's not hard to parameterize solutions to these equations, there are $4$ cases depending on…
Jim
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finitely generated k-algebra and polynomial ring

Let $k$ be a field and let $A \neq 0$ be a finitely generated $k$-algebra, and $x_1, \cdots, x_n$ generate $A$ as a $k$-algebra. Is there any relationship(inclusion, homomorphism, etc.) between $A$ and $k[x_1,\cdots,x_n]$? How about the case if…
Gobi
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Prime spectrum and going-down property

I want to show that $f$ has the going-down property $\Leftrightarrow$ For any prime ideal $\mathfrak{q}$ of $B$, if $\mathfrak{p}=\mathfrak{q}^c$, then $f^{*}:\textrm{Spec}(B_{\mathfrak{q}}) \rightarrow \textrm{Spec}(A_{\mathfrak{p}})$ is…
Gobi
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Deduce that a Noetherian valuation ring is either a field or a Discrete Valuation Ring.

I'm trying to solve this question from a book and I have already proved 1. Let $R$ be a local domain which is not a field. Suppose that the maximal ideal $M$ of $R$ is principal and satisfies $\cap_{n=1}^{\infty}M^n=0$. Show that every non-zero ideal…
Mathproof P.
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Non-zero-divisor in quotient ring: Not a unit over minimal prime?

I think I do need some help with a basic commutative algebra question: Suppose we have a Noetherian integral domain $A$ (we can also assume that $A$ is a $K$-algebra of finite type, though I do not think this is necessary here). Now let $f,g$ be…
johnnycrab
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Disjoint Union of Spectra

The following it from Atiyah-Macdonald's Introduction to Commutative Algebra, exercise 1.22. Apparently this should be very easy, my apologies for asking. I have been stuck for almost 1 day and I cannot figure out what is wrong with my…
Yong Hao Ng
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injective $R$-module homomorphism vs. injective ring homomorphism

The following question has been lingering in my mind for months. Let $R$ be a non-zero commutative ring with $1$. Consider $\phi : R^n \rightarrow R^m$, 1) as an injective $R$-module homomorphism. 2) as an injective ring homomorphism. (by…
Ehsan M. Kermani
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The Jacobson radical of an artinian ring is nilpotent

I am looking for a nice proof for the following lemma, which will later help prove that a ring is artinian if and only if it is noetherian and all prime ideals are maximal. Let $A$ be an artinian ring. Then $\mathfrak{R}$ is nilpotent, id est,…
Berber
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Quotient ring of a localization of a ring

Let $A$ be a commutative ring. Let $P$ be a prime ideal of $A$. Let $I$ be an ideal of $A$ such that $I \subset P$. Let $\bar A = A/I$. Let $\bar P = P/I$. Is $\bar A_{\bar P}$ isomorphic to $A_P/IA_P$?
Makoto Kato
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Does tensoring flat modules preserve minimal generating sets?

Let $k$ be a commutative ring and let $M,N$ be two flat modules over $k$. $\mathbf{EDIT}:$ A minimal generating set $X \subseteq M$ is a set which generates $M$ and no proper subset of $X$ generates $M$. There is no canonical notion of size for a…
Paul Slevin
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