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

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3
votes
3answers
90 views

What can $\operatorname{Hom}\left(\prod_i M_i, N\right)$ look like?

It's easy to see that $\operatorname{Hom}\left(\bigoplus_i M_i, N\right) = \prod_i \operatorname{Hom}(M_i, N)$. However, there are a couple of ways this can conceivably fail if we replace the ...
8
votes
1answer
273 views

Irreducible polynomial over an algebraically closed field

Suppose $k$ is an algebraically closed field and $p(x,y)\in k[x,y]$ is an irreducible polynomial. Prove that there are only finite many $a\in k$ such that $p(x,y)+a$ is reducible, i.e. the set ...
3
votes
1answer
99 views

Local rings and flatness

Let $A \rightarrow B$ be a flat and local homomorphism of commutative local rings. Let $M,N$ be two $B$-modules which are free of finite rank as $A$-modules. Consider the product $M \otimes_B N$ as ...
2
votes
1answer
319 views

Residue field of polynomial rings

Let $k$ be an algebraically closed field of characteristic $p$ and $A=k[x_1,\cdots,x_n]$ the polynomial ring over $k$ in $n$ variables. Given a prime ideal $\mathfrak{p}$ in $A$, denote by ...
2
votes
1answer
235 views

Verifying universal property of Grothendieck group

I'm trying to verify the universal property of the Grothendieck group. Let $\overline{C}$ be the set of isomorphism classes of finitely generated $R$-modules over say a Noetherian ring $R$ and let $C$ ...
4
votes
1answer
145 views

Definition of Grothendieck group

I'm reading the Wiki article about the Grothendieck group. What's the reason we define $[A] - [B] + [C] = 0 $ rather than $[A] + [B] - [C] = 0 $ (or something else) for every exact sequence $0 \to A ...
1
vote
2answers
178 views

Integral dependence and rings of fractions

I have a question on a Proposition in Atiyah and MacDonald's text. It concerns Proposition 5.12 ($A$ and $B$ are commutative rings with an identity) pictured here: Here's my concern: After ...
6
votes
2answers
318 views

Invertible elements in the ring $K[x,y,z,t]/(xy+zt-1)$

I would like to know how big is the set of invertible elements in the ring $$R=K[x,y,z,t]/(xy+zt-1),$$ where $K$ is any field. In particular whether any invertible element is a (edit: scalar) multiple ...
2
votes
1answer
99 views

Technical question on integral ring extensions

Let $A$ be an integral domain, integrally closed in its field of quotients $K$ and let $L$ be a finite Galois extension of $K$ with group $G$. Let $B$ be the integral closure of $A$ in $L$. Let $p$ be ...
2
votes
1answer
371 views

Atiyah-Macdonald, Exercise 2.17 (direct limit)

I have solved the following exercise, can you tell me if this is correct? Thanks. 2.17. Let $(M_i)_{i \in I}$ be a family of submodules of an $A$-module such that for each pair $i,j$ in $I$ there ...
0
votes
1answer
113 views

some question of regular ring

Let $S$ be a polynomial ring with $n$ variables and $I$ be an ideal of $S$. If $S/I$ is regular ring then is $I$ generated by linear forms? is the converse true?
0
votes
1answer
102 views

Graded Ring - Finite Sum

I've just read that if $R=R_0\oplus R_1 \oplus \dots$ is a graded ring and $f\in R$ then there's a unique decomposition of $f$ as $f=f_0+\dots+f_n$ with $f_i\in R_i$. I can't see immediately why in ...
2
votes
1answer
176 views

Integral Galois Extensions - Proposition 2.4 (Lang)

My question refers to the proof of Proposition 2.4, p. 341 in Lang's Algebra. Here is the context: Let $A$ be an integral domain, integrally closed in its field of quotients $K$ and let $L$ be a ...
0
votes
2answers
233 views

Is it Artinian ring?

For a ring $R$ and a graded ideal $I$, let $I_{\geqq p}= \oplus_{i\geqq p}I_i$. If $R/I$ is an artinian, is $R/I_{\geqq p}$ artinian? If it is false, then is it true in polynomial ring case?
2
votes
1answer
148 views

commutativity of torsion functor

For a ring $R$ and finitely generated $R$ modules $U,W$, $${\rm Tor}_i(U,W)={\rm Tor}_i(W,U)$$ for all $i$. I saw proof in Hatcher's book, but I can understand that proof. may be I see another proof? ...
4
votes
0answers
124 views

Is reducedness an open condition?

If $X$ is a (general) scheme and $X$ is reduced at $p$, i.e. $\mathscr{O}_{X,p}$ is reduced, does there necessarily exist an open neighborhood of $p$ on which $X$ is reduced, i.e. $\mathscr{O}_X(U)$ ...
4
votes
1answer
183 views

Is the intersection of two f.g. projective submodules f.g.?

Let $R$ be a commutative unital ring and $M$ a finitely generated projective $R$-module. My question is: if $N_1$ and $N_2$ are f.g. projective submodules of $M$, is $N_1 \cap N_2$ f.g.? Is it ...
2
votes
1answer
190 views

Question about inverse limit

I'm puzzled by the definition of inverse limit in this Wolfram article. I thought if an object was defined by a universal property it meant that the object is unique up to unique isomorphism. This ...
6
votes
3answers
180 views

$I\cdot J$ principal implies $I$ and $J$ principal?

Let $R$ be a Noetherian domain, and let $I$ and $J$ be two ideals of $R$ such that their product $I\cdot J$ is a non-zero principal ideal. Is it true that $I$ and $J$ are principal ideals ? This seems ...
3
votes
2answers
289 views

some exact sequence of ideals

I saw an exact sequence of ideals $$0 \rightarrow I \cap J\rightarrow I \oplus J \rightarrow I + J \rightarrow 0$$In this sequence, maps are ring homomorpism? module homomorphism? And can the above ...
3
votes
2answers
229 views

Proposition 11.4 in Atiyah-MacDonald

I'm not seeing a line in the proof of Proposition 11.4 in Atiyah-MacDonald. Here is a link to some notes I found online which contain the proof: http://folk.uio.no/fredrme/Kommalg.pdf It is also ...
2
votes
3answers
494 views

Zariski topology on prime $\mathrm{Spec}$ of a ring $R$

Let $R$ be a commutative unital ring. Let $\mathrm{Spec}(R) = \{ \mathfrak p \subset R \mid \mathfrak p \text{ a prime ideal of } R \}$. We define a set $C$ to be closed in this space if and only if ...
7
votes
1answer
198 views

Addition and multiplication are continuous in the $I$-adic topology

Can you tell me if this is correct? Let $R$ be a ring and let it have the $I$-adic topology for some ideal $I$ in $R$. I want to show that $+: R \times R \to R$ is continuous at $(x_0, y_0)$. Proof: ...
4
votes
1answer
159 views

basic question on integral schemes

If $X$ is a (reduced) scheme and $P$ is a point of $X$ (not necessarily closed) such that the local ring $\mathcal{O}_{X,P}$ is a regular domain, then must there exist an open affine neighborhood $U = ...
4
votes
1answer
156 views

Flat sheaves over a non-flat base $X \rightarrow Y$

I have a relatively naive question. Suppose that $f: X \rightarrow Y$ is a map of schemes. Then, we get a map of local rings $\mathcal{O}_{Y,f(x)} \rightarrow \mathcal{O}_{X,x}$ and thus for any sheaf ...
3
votes
2answers
159 views

Adic Completion of a direct sum

This is a question related to this. Let $G= \mathbb Z / p \mathbb Z$ for some prime $p$. Let $A = \bigoplus_{n\in \mathbb N} G$, that is, all sequences in $G$ with all but finitely many terms zero. ...
3
votes
1answer
39 views

Quick question about Hausdorffness of completion

Let $G$ be the usual: a topological Abelian group with a topology induced by a countable neighbourhood basis $G_n$ of zero such that $G = G_1 \supset G_2 \supset \dots$. Let $\widehat{G}$ denote the ...
3
votes
1answer
113 views

Quick question about completions and inverse limits

Let $G$ be the usual: a topological Abelian group with a topology induced by a countable neighbourhood basis $G_n$ of zero such that $G = G_1 \supset G_2 \supset \dots$. Let $\widehat{G}$ denote the ...
2
votes
0answers
76 views

First order logic in polynomial equations

Have you ever wondered which points on a conic are the intersections of tangent lines of another surface through the origin? More generally, which points on a shape hold some specified relation to all ...
2
votes
2answers
655 views

What is a lift?

What exactly is a lift? I wanted to prove that for appropriately chosen topological groups $G$ we can show that the completion of $\widehat{G}$ is isomorphic to the inverse limit ...
2
votes
1answer
146 views

Follow up on example computation of $\mathrm{Tor}_n$

I have a follow up question on this question of mine: I can't reconstruct how I got $\operatorname{Im}{d_1^\ast} = 0$ from the following chain: $$0 \to \mathbb Z \otimes_{\mathbb Z} (\mathbb Z / 2 ...
6
votes
1answer
241 views

Are minimal prime ideals in a graded ring graded?

Let $A=\oplus A_i$ be a graded ring. Let $\mathfrak p$ be a minimal prime in $A$. Is $\mathfrak p$ a graded ideal? Intuitively, this means the irreducible components of a projective variety are ...
6
votes
5answers
332 views

Question about proof of $A[X] \otimes_A A[Y] \cong A[X, Y] $

As far as I understand universal properties, one can prove $A[X] \otimes_A A[Y] \cong A[X, Y] $ where $A$ is a commutative unital ring in two ways: (i) by showing that $A[X,Y]$ satisfies the ...
3
votes
2answers
127 views

$B \otimes_A A[X] \cong B[X]$

Let $A$ be a subring of a commutative unital ring $B$. Can you tell me if my proof of the following claim is correct? Claim: $B \otimes_A A[X] \cong B[X]$ Proof: It's enough to show that $B[X]$ ...
2
votes
2answers
572 views

$R/I \otimes_R M \cong M / IM$ [duplicate]

Possible Duplicate: Showing that if $R$ is local and $M$ an $R$-module, then $M \otimes_R (R/\mathfrak m) \cong M / \mathfrak m M$. In one of the answers to one of my previous questions the ...
3
votes
3answers
222 views

Isomorphism from $\widehat{G}$ to $\displaystyle \lim_{\longleftarrow} G/G_n$

Let $G$ be a topological abelian group and let $\widehat{G}$ denote its completion (i.e. equivalence classes of Cauchy sequences). Let $G_n$ be a descending sequence of subgroups, i.e. $G = G_0 ...
6
votes
2answers
709 views

$p$-adic completion of integers

I'm trying to do the following exercise: Let $p$ be a prime and for $n\geq 1$ let $\alpha_n :\mathbb Z/p \mathbb Z \to \mathbb Z/p^n \mathbb Z$ be the injection of abelian groups given by $1 \mapsto ...
6
votes
2answers
1k views

Inverse limit by example

I'm trying to understand inverse limits. For this I am looking at the example (mentioned in Atiyah-Macdonald, page 102): We start with the topological abelian group $G = \mathbb Z$ (endowed with the ...
5
votes
2answers
518 views

Question about $p$-adic numbers and $p$-adic integers

I've been trying to understand what $p$-adic numbers and $p$-adic integers are today. Can you tell me if I have it right? Thanks. Let $p$ be a prime. Then we define the ring of $p$-adic integers to ...
2
votes
1answer
110 views

Localization and Noetherian property

From page 101 in Atiyah-MacDonald: "Two of the important properties of localization are that it preserves exactness and the Noetherian property...." I remember proving that it preserves exactness, ...
3
votes
4answers
121 views

Is it true that if $\alpha \in \operatorname{Frac}(A)$ and $s\alpha \in A$, then $\alpha \in S^{-1}A$?

In the proof of Proposition 1.9 in Chapter VII of Algebra by Serge Lang, it seems to me that the following property is used. Let $A$ be a commutative entire ring, $S$ a multiplicative subset of $A$, ...
5
votes
2answers
915 views

Classification of prime ideals of $\mathbb{Z}[X]/(f(X))$

Let $\mathbb{Z}[X]$ be the ring of polynomials in one variable. Let $f(X) \in \mathbb{Z}[X]$ be a monic irreducible polynomial. Let $A = \mathbb{Z}[X]/(f(X))$. Let $\theta$ = $X$ (mod $f(X)$). My ...
33
votes
1answer
3k views

Classification of prime ideals of $\mathbb{Z}[X]$

Let $\mathbb{Z}[X]$ be the ring of polynomials in one variable. My question: Is every prime ideal of $\mathbb{Z}[X]$ one of following types? If yes, how would you prove this? (1) $(0)$ (2) ...
2
votes
1answer
200 views

Question about the $\mathrm{Tor}$ functor

Assume we want to define $\mathrm{Tor}_n (M,N)$ where $M,N$ are $R$-modules and $R$ is a commutative unital ring. We take a projective resolution of $M$: $$ \dots \to P_1 \to P_0 \to M \to 0$$ Now ...
5
votes
2answers
171 views

Is $\mathbb{Z}[\sqrt{2},\sqrt{3}]$ flat over $\mathbb{Z}[\sqrt{2}]$?

Is $\mathbb{Z}[\sqrt{2},\sqrt{3}]$ flat over $\mathbb{Z}[\sqrt{2}]$? The definitions doesn't seem to help. An idea of how to look at such problems would be helpful.
4
votes
2answers
120 views

Proper 2-generator ideal is not the intersection of 2 proper coprime ideals

Let $R$ be a polynomial ring (in finitely many, say 8, indeterminates) over an algebraically closed field $k$. Suppose $I=(f,g) \neq R$ is a proper ideal of $R$ and $I = J \cap L$ for two proper ...
0
votes
1answer
42 views

False proof of $R$ Noetherian, $I$ irreducible hence $I$ prime

Can you tell me what's wrong with my proof? Thanks. Claim: If $R$ is a Noetherian ring and $I$ is an irreducible ideal in $R$ then $I$ is prime Proof: Let $xy \in I$. We want to show that either ...
1
vote
1answer
115 views

Question about primary decomposition in Noetherian rings

I have a question about the following proof: How do I get that $\mathfrak a$ is reducible? I thought perhaps one can argue that $\mathfrak a \cap \mathfrak a = \mathfrak a$ is a finite intersection ...
21
votes
2answers
739 views

The prime spectrum of a Dedekind Domain

Let $A$ be a Dedekind Domain, let $X = \operatorname{Spec}(A)$. Are all open sets in $X$ basic open sets? Thinking about the Zariski topology (in the classical sense) of a non-singular affine curve, ...
1
vote
2answers
93 views

Question about factor rings

Assume $m_i$ are maximal ideals in a ring $R$. Then I have $m_1 \cdot \dots m_{k}$ is an ideal in $m_1 \cdot \dots m_{k-1}$ hence I can quotient to get a factor ring $m_1 \cdot \dots m_{k-1} / m_1 ...