# Tagged Questions

Questions about commutative rings, their ideals, and their modules.

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### Atiyah and MacDonald Theorem 9.5

$K$ is an algebraic number field, $A$ its ring of integers. Theorem 5.17 shows that $A\subseteq\sum\mathbb{Z}v_j$ with $v_j\in K$. Theorem 9.5 then concludes that $A$ is a f.g. $\mathbb{Z}$-module. I ...
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### Depth of a module over local ring and vanishing of Ext functor

I'm studying depth of $A$-modules, where $A$ is a noetherian ring, in Matsumura's Commutative Algebra text and I'm experiencing some trouble understanding the proof of a basic result. I think all of ...
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### Definition of free module

i) Let $M$ be a free $R$-module. By definition $M = R \oplus R \oplus\cdots\oplus R$ . Can anyone could explain me why $M = Rx_1 \oplus\cdots\oplus Rx_n$ where $x_1,\ldots,x_n$ elements of $M$. My ...
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### If $J$ is the ideal generated by all idempotents in a prime ideal, then $R/J$ has only trivial idempotents

Let $R$ be a commutative ring with identity, $P$ be a prime ideal in $R$ and define $$X := \lbrace t \in P \mid t^2=t \rbrace.$$ Also let $J$ denote the smallest ideal of $R$ that contains $X$. ...
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### Find shortest primary decomposition.

Let $A=k[x,y,z]$ and let $T_1=(x,y)$, $T_2=(x,z)$. Define $I=T_1T_2$ and calculate the shortest primary decomposition of $I$. I dont know where to start and I am looking for hints, how should I think ...
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### Show that $(A / \mathfrak{a}) \otimes_A M \cong M / \mathfrak{a} M$ for a ring $A$, ideal $\mathfrak{a}$, $A$-module $M$.

This is Atiyah-Macdonald Exercise 2.2 Exercise: Let $A$ be a ring, $\mathfrak a$ an ideal, $M$ an $A$-module. Show that $(A/\mathfrak a) \otimes_A M$ is isomorphic to $M/\mathfrak aM$. [Tensor the ...
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### Nontrivial example of an artin algebra R such that R is pure-injective as an R-module

Give a nontrivial example of an artin algebra $R$ such that $R$ is pure-injective as an $R$-module. Clearly $0$-Gorenstein (self-injective) artin algebra has this property. Can anyone give me a ...
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### When a prime ideal is restricted to a basic open subset of projective space, is it still prime?

Suppose $I\subset k[x_0,\ldots,x_n]$ is a prime ideal. Now restricted on the basic open subset $\mathbb{P}^n_{x_i}$ of $\mathbb{P}^n$, is $I$ still prime? Note: 1. Here $\mathbb{P}^n_{x_i}$ is ...
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### $E \to S$ surjective in degrees $\geq 1$ implies $\widetilde{E} \to \widetilde{S}$ surjective

In the proof of Theorem II.8.13 in Hartshorne (which is the Euler sequence), the author writes: Let $S = A[x_0, \ldots, x_n]$. [...] The exact sequence $$0 \to M \to E \to S$$ of graded $S$-...
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### Restriction and extension of scalars between flat algebras and their completion over a DVR and ideals.

So, in a proof I am currently reading I have stumbled upon the following. Let $R$ be a discrete valuation ring, $\hat{R}$ its completion and $t$ a uniformizing parameter for $R.$ Let $A$ be a flat $R$...
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### pictorial illustration of simplicial complexes

Consider the following two complexes (Bruns&Herzog p.215): By just looking at the complex on the left, i am not sure how to read its faces. Surely its vertices are $v_1,v_2,v_3,v_4,v_5$. The ...
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### Invertible matrices in commutative rings

Let $A$ be a square matrix over a commutative ring $R$. Then $A$ has a left inverse iff it is invertible. Does there exist a elementary proof of this fact? (i.e. without using the determinant!)
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### local PID that is not a field is a DVR

I would be very happy if someone would check my proof of the fact that a local PID that is not a field is a DVR: Let $A$ be a local PID that is not a field. Since irreducibles generate maximal ideals ...
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### Direct-Sum Decomposition of an Artinian module

Let $R$ be a commutative Noetherian ring. Suppose $M$ is a finitely-generated non-zero Artinian $R$-module. Question: How can we prove that there are maximal ideals $m_1 , m_2 , \ldots , m_n$ such ...
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### Find the field of fractions and the integral closure of a subring of $\mathbb Z[x]$.

Let $R$ be a subring of $\mathbb{Z}[x]$ consisting of polynomials such that the coefficients of $x$ and $x^2$ are zero. Find the field of fractions of $R$. Find the integral closure of $R$ in it's ...
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### Can $\operatorname{Spec}(A)$ be expressed as an inverse limit?

We know that given a ring $A$ such that $A/\mathfrak{R}$ is absolutely flat, then $\operatorname{Spec}(A)$ is Hausdorff (it's an equivalence). So $Spec(A)$ becomes a quasi-compact, Hausdorff and ...