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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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I have trouble understanding this statement about (basic) commutative algebra

Any homomorphic image of a Noetherian ring is Noetherian. Furthermore, if $R_0$ is a Noetherian ring, and $R$ is a finitely generated algebra over $R_0$, then $R$ is Noetherian. Noetherian is a definition ascribed for a ring. But in the second…
jk001
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Meaning of $(\mathfrak q :x)$ in ring theory (commutative algebra)

I see the following notation: for a ring $A,$ a prime ideal $\mathfrak q,$ and $x\in A$. $$ (\mathfrak q :x) = (1). $$ The question is: what does $(\mathfrak q :x)$ mean? The ideal generated by $\mathfrak q ,x?$ Then why do we need to use a colon?
Ma Joad
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Integrality of elements in the Field of Fractions

Let $R$ be a domain with field of fractions $K$ and suppose that $t$ is a transcendental element over $K$. Consider $f(t)=\sum_{k=0}^l a_k t^k,\ \in K[t]$. Show that if $f(t)$ is integral over $R[t]$, then $a_k$ is integral over $R$ for each $k$. My…
Edward Teach
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Proposition 9.7 of Atiyah

Let $A$ be an integral domain, $K$ its field of fractions. An $A$-submodule $M$ of $K$ is a fractional ideal of $A$ if $xM\subset A$ for some $x\neq 0$ in $A$. Proposition 9.7. Let $A$ be a local domain. Then $A$ is a discrete valuation ring $\iff$…
Zeldovich Yakov
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Question about definition of Spec function

If we have commutative rings $R$ and $T$ and a homomorphism between them $\phi : R \to T$, then we define $Spec(\phi): Spec(T) \to Spec(R)$ by $p \mapsto \phi^{-1}(p)$. There is then a theorem in the notes that I am reading that states that if…
Orlly
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Equivalence relation of having a zero divisor in localisation

If we take a commutative unital ring $R$ and a multiplicative subset $S$, we localise at $S$ using the equivalence relation on $R \times S$ given by $(a,s) \sim (b,t)$ if and only if there is $u \in S$, such that $u(ta-sb)=0$. As far as I can tell,…
Orlly
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Proof of radicals of ideals being equal

I am reading some Commutative Algebra notes, and have come across the following result: Let $R$ be a commutative unital ring, and let $a$ and $b$ be ideals of R. Then we have $(a \cap b) \cdot (a \cap b) \subseteq a \cdot b \subseteq a \cap b$ and…
Orlly
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Does $P\subset Q$ imply $A_P \subset A_Q$?

P and Q are prime ideals of A such that $P\subset Q$ Whether $A_P \subset A_Q$ or $A_Q\subset A_P$? ($A_P$ and $A_Q$ are localization at P and Q respectively ) I think it will be $A_Q\subset A_P$. As $P\subset Q$ will imply $P^c \supset Q^c$, so…
simu tiyam
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Can every element of the completion $\widehat{A}$ of a local ring $A$ be written as an element of $A$ with a unit of $\widehat{A}$?

Let $A$ be a noetherian local ring, $\widehat{A}$ its completion. Can every element $f\in\widehat{A}$ be written as a product $f'u$ with $f'\in A$ and $u\in\widehat{A}^\times$?
stupid_question_bot
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Why is the power of maximal ideals comaximal in an Artinian ring?

Let $R$ be an Artinian ring, commutative with 1. We know : there are only finitely many maximal ideals of $R$. $Jac(R)^m = 0$ for some natural number $m$, using D.C.C. Every prime ideal is maximal, using 2. The backdrop of this question is we want…
Jun Xu
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Corollary 1.9 in Eisenbud's Commutative Algebra

(See statement and proof attached below.) In his proof of Corollary 1.9, he states that $\mathfrak{m}_p \supset I(X)$ iff $p\in X$. I can see one direction, namely if $p\in X$ then $\mathfrak{m}_p \supset I(X)$. Why is the other direction true? I…
klein4
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Every prime in a graded ring is nearly homogeneous?

Assume that $S_0$ is a field and that $S$ is a graded ring generated as an $S_0$-algebra by $S_1$. Let $Q\subset S$ be a prime ideal, and let $P\subset Q$ be the ideal generated by the set of homogeneous elements of $Q$. Then, it's said that $P$ is…
Yuyi Zhang
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Formal argument for why $(\mathbb{Z}/(4))_{(2)} = \mathbb{Z}/(4)$

I'm learning about localizations, and I came across this statement: $$ \left(\mathbb{Z}/(4)\right)_{(2)} = \mathbb{Z}/(4) $$ Now, this statement makes sense because inverting all of the odd elements have no effect since the images of odd elements in…
klein4
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Residue field of embedded ring

Suppose I have an embedding $R\rightarrow S$ where $R$ is a subring of $S$. Given a prime ideal $\mathfrak{p}\subset S$ and its preimage $\mathfrak{q}\subset R$, is it true that $\kappa(\mathfrak{q})\subseteq\kappa(\mathfrak{p})$?
Christina
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Units in Tensor Algebra

Let $M$ be an $R$-module, and let $T(M) = \bigoplus_{k \geq 0 } T^k(M)$ denote the tensor algebra of $M$. By definition $T^0(M)=R$ . I am looking for a simple explanation to this problem. I am looking for units in $T(M)$. I can figure out that the…
zero2infinity
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