# Tagged Questions

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

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### Derived functors of torsion functor

Let $A$ be a domain. For every $A$-module $M$ consider its torsion submodule $M^{tor}$ made up of elements of $M$ which are annihilated by a non zero-element of $A$. If $f \colon M \to N$ is a ...
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### Zero divisors, nilpotents and units in the ring of functions $\mathbb{R} \to \mathbb{R}$

Let $R$ be the set of all real valued functions defined for all real numbers under function addition and multiplication. i have to show that all the zero divisors of $R$ all nilpotent elements of $R$...
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### Non-zero divisor in an integral domain

Let $R$ be an integral domain and $P$ a prime ideal. Let $x$ be an element such that $xP^{m-1}=P^m$ for some $m>0$. Is $P$ generated by $x$?
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### Tensor products of infinite-dimensional spaces and other objects

It has just occurred to me that most of my intuition for tensor products is derived from the special case of finite-dimensional vector spaces, so I'm wondering which properties I've taken for granted ...
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### A tensor product of power series

Let $k$ be a field. I am wondering if there is an easy description of the ring $$k[[x]] \otimes_{k[x]} k[[x]]$$ that is the tensor product of the power series ring $k[[x]]$ with itself over the ring ...
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### A finite commutative ring with the property that every element can be written as product of two elements is unital

I was struggling for days with this nice problem: Let $A$ be a finite commutative ring such that every element of $A$ can be written as product of two elements of $A$. Show that $A$ has a ...
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### The bijection between homogeneous prime ideals of $S_f$ and prime ideals of $(S_f)_0$

It is well-known that if $S$ is a graded ring, and $f$ is a homogeneous element of positive degree, then there is a bijection between the homogeneous prime ideals of the localization $S_f$ and the ...
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### Are the determinantal ideals prime?

I want to prove the determinantal ideals over a field are prime ideals. To be concrete: For simplicity, let $I=(x_{11}x_{22}-x_{12}x_{21},x_{11}x_{23}-x_{13}x_{21},x_{12}x_{23}-x_{13}x_{22})$ be ...
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### Proof that a certain derivation is well defined

I have spent several hours on this, apparently straightforward issue. This is with reference to page 17 in the following notes http://www.math.lsa.umich.edu/~hochster/615W10/615.pdf Suppose, $R$ is ...
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### Every set of $n$ generators of $A^{n}$ is actually a basis

Let $A$ be a commutative ring with $1$. It is a standard result that every set of $n$ generators of the free $A$-module $A^{n}$ is actually a basis. The proof uses tensor products. I was reading a "...
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### Regularity ascends from a Noetherian ring to a polynomial or power series ring over it

I am looking for a proof of the following statement: A Noetherian ring $R$ is regular if and only if $R[x]$ is regular if and only if $R[[x]]$ is regular. I am trying to understand the properties ...
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### Integral extensions: one prime lying over implies equal localization

Here's a problem from Matsumura's book "Commutative ring theory" page $69$. Let $A$ be a ring and let $A \subset B$ be an integral extension, and $\mathfrak{p}$ a prime ideal of $A$. Suppose that $B$ ...
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### Grothendieck spectral sequence

given functors $F,G$, left exact, with as good properties as you want we have a spectral sequence $R^p F\circ R^q G$ abutting to $R^{p+q}(F\circ G)$. I am looking for an analogous for a "mixed version"...
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### Dimensions of modules of the maximal compact subrings of locally compact fields

I have checked the list of similar titles, proposed by the site. I hope this is not a repetition. This question arises from a proof of a proposition in the book Basic Number Theory, as follows. ...
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### Algebra over a ring

Could someone point me to a proof which shows that an algebra over a ring can be presented as a quotient of a polynomial ring (in possibly infinitely many variables).
Let $A$ be a commutative ring with $1$. Suppose that $P \subseteq Q$ are prime ideals in $A$ and that $M$ is an $A$-module. Prove that the localization of the $A$-module $M_{Q}$ at $P$ is the ...
If there is a ring homomorphism $A\rightarrow B$ and if $Q$ is an injective $A$-module, is it true that $Q\otimes_A B$ is an injective $B$-module? I don't think it's true but can't think of a ...