# Tagged Questions

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

159 views

### Does it hold that the $p$-adic completion of the integers equals the completion of the localization in $p$?

maybe this is a stupid question, but I could not solve it even for the ordinary integers $\mathbb{Z}$. Furthermore, I don't have to much knowledge on algebraic number theory and ramifications. Let $K$...
317 views

### Universal property of polynomial ring in $\mathbf{CRING}$

I know that the polynomial ring $A[x]$ is the free $A$-algebra on $\{x\}$; this is its universal property in the category of $A$-algebras. Is there also a universal property for $A[x]$ considered as a ...
98 views

42 views

### Equivalent definitions of fractional ideals

Let $R$ be an integral domain and $K$ its field of fractions. The usual definition of fractional ideal $I$ ($I$ is an $R$-submodule of $K$) is that for some nonzero $r\in R$ we have $rI\subset R$, and ...
343 views

### Finitely generated ideal in Boolean ring; how do we motivate the generator?

This problem is Exercise 11.3 in Atiyah/Macdonald Commutative Algebra. They ask to prove every finitely generated ideal in a Boolean ring is in fact a principal ideal. The question has been answered ...
205 views

### Atiyah-Macdonald 2.3

In solving question 2.3 from Atiyah & Macdonald's commutative algebra textbook, I run into the following difficulty: Let $A$ be a local ring with $k:= A/mA$ its residue field and let $M$ and $N$ ...
114 views

### Some questions on Hartshorne I.7: intersections in projective space

I am reading I.7 of Hartshorne, and here are some questions I don't understand. 1) Prop. 7.4. Let $M$ be a finitely generated graded module over a noetherian graded ring $S$. Then there exists a ...
69 views

### A Direct Sum of Members of a Certain Class of Modules

Let $S$ be a class of $R$-modules and let an $R$-module $M$ be countably generated. Suppose that, for every direct summand $K$ of $M$, each element of $K$ belongs to a direct summand of $K$ that is ...
204 views

### What are local homomorphisms, geometrically?

For want of a better name, let us say that a ring homomorphism $f : A \to B$ is local if it (preserves and) reflects invertibility, i.e. $f (a)$ is invertible in $B$ (if and) only if $a$ is invertible ...
51 views

### Is the colimit of finite tensor products a tensor product?

Let $(R_\lambda)_{\lambda\in\Lambda}$ be a family of $A$-algebras. Atiyah & MacDonald defines the "tensor product" of the family as the direct limit of the tensor product of finite subfamilies. ...
158 views

### Integral closure of 1-dimensional noetherian local domains

Let $(R,m)$ be a $1$-dimensional noetherian local domain and $S$ its integral closure. Clearly $S$ is $1$-dimensional noetherian semi-local domain. Is $mS=J(S)$, where $J(S)$ is the Jacobson radical ...
239 views

### Surjectivity implies injectivity of finitely generated modules, localization?

The following problem is canonical: Suppose $A$ is a commutative unitary ring, and $M$ is a finitely generated module over $A$. If an endomorphism $f\colon M\to M$ is surjective, then it's also ...
63 views

### Contracted ideals in number fields

I am trying to translate a section of Wolfgang Krull's report "Idealtheorie". At one point (Section $7$ on Quotient Rings) I believe that he makes something like the following statement: Suppose for ...
185 views

### Is every affine scheme the complement of the closed point $x$ of the spectrum of a local ring $A$?

Let $R$ be a commutative ring with identity element and let $\operatorname{Spec}(R)$ be the associated affine scheme. Does for each affine scheme $\operatorname{Spec}(R)$ exist a local ring $A$ ...
80 views

### Isomorphism of modules arising from algebraic topology

While studying for a course in algebraic topology, the following question popped out: Let $S,R$ be two commutative rings with unit, $A,B$ two $S$-modules, and assume that $R$ is also an $S$-module....
150 views

### Question on calculating hypercohomology

I want to compute the algebraic de Rham cohomology of $\mathbb{C}^*$, and I'm confused. I don't have much background in this, so I was hoping a very concrete example would clear up a lot of this ...
144 views

### Factorization of ideals in a coordinate ring (Dedekind domain)

Consider $f \in \mathbb{C}[X,Y]$ an irreducible curve non singular. Let $A = \mathbb{C}[X,Y] / (f)$ be the coordinate ring of $f$ and choose a curve $g \in \mathbb{C}[X,Y]$ with no component in common ...
45 views

114 views

### Ring such that $q^2\mid p^2$ does not imply $q\mid p$?

Let $R$ be a commutative ring with $1$ and suppose $q^2\mid p^2,$ for $p,q \in R$. Unless $R$ is a UFD, I don't believe I can conclude that $q\mid p,$ but I would like to know a concrete ...
129 views

### Is every local ring the localization of some other ring?

One way of constructing a local ring is to start with any commutative ring, and localize all the elements outside of some maximal ideal (i.e., adjoining inverses to all those elements). But I'm ...
241 views

### Euclid's proof of the infinitude of primes to prove this question

I'm trying to prove that if $k$ is a field, then there are an infinite number of irreducible monic polynomials in $k[X]$. My attempt of solution is use almost the same strategy of the Euclid's proof ...
126 views

### Irreducibility of some multivariate polynomials

Consider the polynomials $xw-yz\in A[x,y,z,w]$ and $x^n+y^n+z^n\in A[x,y,z]$, where $A$ is a commutative ring. I am curious to know what conditions on $A$ (factorial ring, algebraically closed field, ...
94 views

### Zorn's lemma and maximal ideals

Let's consider two statements: Zorn's lemma and theorem about existence of maximal ideals in commutative ring with $1$. It's easy to prove that Zorn's lemma implies existence of maximal ideals. I ...
145 views

### Submodules and quotients of free modules over Noetherian local rings

Let $R$ be a Noetherian local commutative ring, $F$ a finitely generated free $R$-module and $A,B$ some arbitrary $R$-modules. Consider a short exact sequence $0 \to A \to F \to B \to 0$. In [Bruns, ...
227 views

### Difference between algebraic and integral extension

I have been reading Miles Reid Undergraduate Commutative Algebra and in chapter 4 he talks about a crucial difference between algebraic extension and integral extension (see the picture below). Now I ...
120 views

### Strong approximation theorem for Dedekind Domains

This is a theorem in "Maximal Orders" by Reiner. Page 48 stated without proof. And is said to be an easy consequence of The Chinese remainder Theorem. I am attempting to prove the theorem and need a ...
412 views

### Every Artinian ring is isomorphic to a finite direct product of Artinian local rings

I was reading a proof of the above theorem (1.6.7 Theorem) from here, but there was something that confused me. The proof says $R$ has finitely many maximal ideals $M_1, \ldots ,M_r$, and the ...
147 views

### Kähler differentials of the cuspidal cubic

I want to compute $\Omega^1_{A,\mathbb{C}}$ for $A = \mathbb{C}[X,Y]/(Y^2 - X^3)$, or more precisely, I want to show that the module of Kähler differentials is free of rank 2 at the origin, and free ...
103 views

### Example of strict inclusion for the localization of associated primes

Let $A$ be a commutative ring and $M$ an $A$-module. It is well known that $$\operatorname{Ass} M\cap\operatorname{Spec}S^{-1}A\subset\operatorname{Ass}S^{-1}M,$$ and that equality holds if $A$ (or $M$...