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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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Question about homomorphisms $f_{!}, f^{!}$.

Let $f: A \to B$ be a finite ring homomorphism and $N$ a $B$-module. $N$ can be considered as an $A$-module if we define $A \times N \to N$, $(a, n) \mapsto f(a)n$. Therefore we have a map $f_{!}: K(B) \to K(A)$. Let $M$ be a $B$ module. $B$ can be…
LJR
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Question about radical of powers of prime ideals.

Let $Q$ be an ideal of a commutative ring $A$ and $$r(Q) = \{x \in A : x^n \in Q \text{ for some } n >0 \},$$ the radical of $Q$. Suppose that $P$ is a prime ideal of $A$. How to show that $r(P^n) = P$ for all $n>0$? It is clear that $P \subseteq…
LJR
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A regular system of parameters of a ramified complete regular local ring

Let $(R,\mathfrak m)$ be a ramified complete regular local ring of dimension $n$ with $\operatorname{char} R/\mathfrak m=p$. Then $\operatorname{ht}(pR)=1$ and the question is: How to choose a regular system of parameters $x_1,\dots,x_n$ of $R$…
nick
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Atiyah-Macdonald: Exercise 3.6

Q: Let $A$ be a nonzero ring and let $\Sigma$ be the set of multiplicatively closed subsets of $A$ such that $0 \notin S.$ Show that $\Sigma$ has a maximal element, and that $S \in \Sigma$ is maximal iff $A \setminus S$ is a minimal prime ideal of…
Saikat
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Integral closure of k-algebra

Let $k$ be a field and $A$ a finitely generated algebra over $k$ that doesn't have zero divisors. Why is the integral closure of $A$ a finitely generated module over $A$ ? (edited)
Radu Titiu
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Every $2$-dimensional commutative $k$-algebra with only one prime ideal is isomorphic to $k[x]/(x^2)$

Let $k$ an algebraically closed field. I want to prove the following: Every $2$-dimensional $k$-algebra with only one prime ideal is isomorphic to $k[x]/(x^2)$. Not every $3$-dimensional $k$-algebra with only one prime ideal is isomorphic to…
LeviathanTheEsper
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Field of algebraic numbers over Q with p-adic value

Define $\overline{\mathbb{Q}} \subset \mathbb{C}$ to be the subset consisting of all complex numbers which are algebraic over $\mathbb{Q}$. We know that $\overline{\mathbb{Q}}$ is a countable field and that is algebraically closed. Show that there…
TJIF
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Showing surjectivity of trace map $B\to A$ from faithful flatness

I've come across the claim if $A\hookrightarrow B$ is a finite etale extension of rings (commutative w/ $1$) with $A$ Noetherian then the trace map $\operatorname{Tr}_{B/A}:B\to A$ is surjective and am looking for some help to see why this is…
Alex Mathers
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about the definition of extended ideals

Let $A, B$ be commutative rings with multiplicative identity, let $f: A \rightarrow B$ be a ring homomorphism, let $\mathfrak{a}$ be an ideal in $A$. The extension $\mathfrak{a}^\text{e}$ of $\mathfrak{a}$ in $B$ is defined to be the ideal…
Tom Jonathan
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Localization over commutative Noetherian rings

Let $S$ be a multiplicatively closed subset of a commutative noetherian ring $A$. Let $M$ and $N$ be finitely generated $A$-modules. If $M_S$ is isomorphic to $N_S$, show that $M_t$ is isomorphic to $N_t$ for some $t \in S.$
user72578
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Localization of $\mathbb{C}[x,y]/(x^{3}-y^{3})$

Consider the ring $R=\mathbb{C}[x,y]/(x^{3}-y^{3})$ and let $S$ be the set of all non-zero divisors of $R$. How to find $S^{-1}A$? I guess the idea is to find a ring which is isomorphic to (or perhaps that contains a subring isomorphic to…
user6495
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Isomorphism or non-isomorphism of two specific local rings

Let $K$ be a field and set $A=K[X,Y]/(XY)$ and $B=K[X,Y]/(Y^2-X^3-X^2)$. Are the two local rings $A_{(X,Y)}$ and $B_{(X,Y)}$ isomorphic? I think that they are non-isomorphic but I can't prove this.
Ronald Bernard
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Atiyah Macdonald 3.4

I'm a little confused about the following exercise in A-M: Let $f:A\to B$ be a homomorphism of rings and let $S$ be a multiplicatively closed subset of $A$. Let $T=f(S)$. Show that $S^{-1}B$ and $T^{-1}B$ are isomorphic as $S^{-1}A$-modules. You…
Alvin Chen
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Counterexample to claim that Jacobson radical of a commutative ring is given by checking against units

Let $R$ be a commutative ring and $J(R)$ its Jacobson radical. It is easy to check that $J(R)$ consists of exactly those $f$ for which $1 + gf \in R^\times$ for all $g$. Let $J'(R)$ be those $f$ for which $1 + uf \in R^\times$ for all units $u$. We…
forget this
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Is the localisation of $\mathbb{Z}[X,Y]/(XY-9)$ to $\mathfrak{m} = (x,y,3)$ regular?

Let $A$ be the localisation of $R = \mathbb{Z}[X,Y]/(XY-9)$ at $\mathfrak{m} = (x,y,3)$. I wondered if this ring is maybe Noetherian, Artin, or regular. Noetherian: Notice that $\mathbb{Z}$ is a PID, and thus Noetherian. By Hilberts' basis theorem…
JanBakfiets1
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