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

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2
votes
2answers
167 views

For a ring R with an ideal I, the I-adic topology makes R into a topological ring

Let $R$ be a commutative ring with identity. Let $I$ be an ideal of $R$. Suppose, we give a topology on $R$ where a set is open if and only if it is a union of cosets of powers of $I$. Then, is $R$ a ...
5
votes
1answer
100 views

Locally a domain and connected implies a domain

Let $R$ be a commutative ring with unit. Let $R_p$ be a domain for all $p\in SpecR$ and let $SpecR$ be connected. Is it true that $R$ is a domain or can someone provide a counterexample. Note here ...
8
votes
1answer
234 views

Why is UFD a Krull domain?

Matsumura mentions this as if it is obvious, and I can't find this result anywhere. Am I missing something obvious here?
12
votes
2answers
350 views

Does inclusion of a ring into a polynomial ring induce a closed map on prime spectra?

Let $A$ be a commutative (unital) ring, and $A[x_1,\ldots,x_n]$ a polynomial ring over it in some finite number of variables. The inclusion $i\colon A \hookrightarrow A[x_1,\ldots,x_n]$ induces (by ...
3
votes
1answer
131 views

Find two submodules $E_1$ and $E_2$ such that $\textrm{Ass}(E_1) \cup \textrm{Ass}(E_2) \subsetneq \textrm{Ass}(E_1 + E_2)$

I'm looking for a module $E$ of a commutative ring $A$ with two submodules $E_1$ and $E_2$ such that the associated primes of $E_1 + E_2$ strictly contain the union of the associated primes of $E_1$ ...
2
votes
2answers
151 views

Local complete intersections and the cotangent module

Let $A$ be a (commutative) noetherian ring, and let $I \subseteq A$ be an ideal. It is not hard to show that if $I$ is generated by a length $n$ regular sequence, then the $A/I$-module $I/I^2$ is a ...
2
votes
1answer
137 views

Subsequences of regular sequence

I was trying to answer this question - whether a subsequence of a regular sequence is regular in a Noetherian ring which is not local. In the local case, regular sequences can be permuted and so a ...
10
votes
1answer
287 views

Minimal spectrum of a commutative ring

Can anyone explain to me why the minimal prime ideals of a commutative ring (with the subspace topology inherited from the Zariski topology) form a totally disconnected space, or give a reference? I ...
2
votes
2answers
461 views

Example of an injective module

I can't find an example of a countable injective module over a non-Noetherian ring.
6
votes
3answers
521 views

Why is the prime spectrum of a domain irreducible in the Zariski topology

How does one show that the prime spectrum of a domain is irreducible in the Zariski topology?
2
votes
1answer
447 views

Max Noether's $AF + BG$ theorem

Wikipedia tells me about Max Noether's $AF + BG$ theorem but only gives one reference and one external link. I've had a look at the MathWorld link but it seems to be an entirely geometric formulation ...
10
votes
1answer
405 views

Finitely generated modules over PID

Let $A$, $B$, $C$, and $D$ be finitely generated modules over a PID $R$ such that $A\oplus $ $B$ $\cong$ $C\oplus $ $D$ and $A\oplus $ $D$ $\cong$ $C\oplus $ $B$ . Prove that $A$ $\cong$ $C$ and $B$ ...
2
votes
2answers
164 views

why a reduced ring (commutative with 1) can be embedded into a sum of integral rings?

the question is exactly "why a reduced ring (commutative with 1) can be embedded into a sum of integral rings?" Is this simply because in the normalization process we can have many irreducible ...
5
votes
1answer
96 views

In what generality does the following isomorphism involving tensors and homs hold?

Let $R$ be a CRing and let $M,N$ be $R$-modules. Let $M^*:=Hom_R(M,R)$. I have seen the following isomorphism asserted in the case where $R$ is a field and $M$ and $N$ are f.g. vector spaces: ...
1
vote
1answer
133 views

In what generality does the second argument of Hom distribute over tensor?

Let $R$ be a commutative ring, and let $M,N,P$ be $R$-modules. In what generality can we say that $Hom_R(M,N\otimes_R P)\cong Hom_R(M,N)\otimes_R Hom_R(M,P)$. This is true in a cartesian monoidal ...
2
votes
1answer
280 views

Tensoring a monomorphism of free modules with an identity map

Suppose $R$ is a commutative ring, $f\colon F_1\to F_2$ is a homomorphism of free modules, and $M$ is an $R$-module. If $f$ is a surjective homomorphism, then $f\otimes_R \mathrm{id}_M$ is ...
12
votes
1answer
1k views

Given a commutative ring $R$ and an epimorphism $R^m \to R^n$ is then $m \geq n$?

If $\varphi:R^{m}\to R^{n}$ is an epimorphism of free modules over a commutative ring, does it follow that $m \geq n$? This is obviously true for vector spaces over a field, but how would one show ...
8
votes
2answers
286 views

Is Nullstellensatz true for arbitrary fields if there aren't hidden points?

The ideals $I=(X,Y)$ and $J=(X^2+Y^2)$ in $\mathbb R[X,Y]$ are such that $V(I)=V(J)$ and their radicals aren't the same contradicting the Nullstellensatz (in case it was true for arbitrary fields). ...
3
votes
1answer
120 views

elementary k-automorphism of $k[x_1, \ldots, x_n]$

We define the following subgroups of the automorphism group of $k[x_1, \ldots, x_n]$ E(k,n) is the subgroup generated by elementary $k$-automorphisms of $k[x_1, \cdots, x_n]$ of the form $(x_1, x_2, ...
13
votes
3answers
1k views

Homomorphisms of graded modules

Let $M$ and $N$ be graded $R$-modules (with $R$ a graded ring). $\varphi:M\rightarrow N$ is a homogeneous homomorphism of degree $i$ if $\varphi(M_n)\subset N_{n+i}$. Denote by $\mathrm{Hom}_i(M,N)$ ...
3
votes
3answers
247 views

Hausdorffness of a topological abelian group

In the chapter on completions in Atiyah Macdonald, they define a topological abelian group. Let $G$ be such a group. Denote $H$ to be the set of intersection of neighbourhoods of $0$. Then it is shown ...
8
votes
2answers
604 views

Can a quotient field ever be finitely generated as an algebra?

If A is a commutative integral domain that's not a field, and let $K$ be the quotient field of A. We know that $K$ is not finitely generated as an A-module. But can $K$ ever be finitely generated as ...
2
votes
0answers
377 views

Completion of a polynomial ring w.r.t the homogeneous maximal ideal is the power series ring

I am trying to understand the proof of this fact. On page 183, Eisenbud defines a map from the formal power series ring $S[[x_1,x_2,...,x_n]]$ to $R/m^i$ where $R=S[x_1,x_2,...,x_n]$ sending $f$ to ...
8
votes
2answers
609 views

If A is noetherian, then Spec(A) is noetherian

Let A be a noetherian ring. How can I show that Spec(A) is noetherian? Also, is there a way to show this by showing directly that the closed sets in Spec(A) satisfy the descending chain condition? ...
0
votes
2answers
651 views

Is a finitely generated torsion module over PID an injective module?

Since I often draw wrong conclusions, I need a confirmation. My argument is that for nonzero prime $\mathfrak{p}\subset A$, $A/\mathfrak{p}$ is injective being a field (hence divisible), and $A/p^{k}$ ...
3
votes
0answers
157 views

symmetric algebra

Suppose we have the following situation: $I = \oplus_{i=1}^{p-1}I_1$ is a decomposition of an $R$-ideal in invertible modules of rank $1$, $I_iI_j \subset I_{i+j \pmod{p-1}}$ and $I_1^i = I_{i ...
10
votes
2answers
1k views

When is a tensor product of two commutative rings noetherian?

In particular, I'm told if $k$ is commutative (ring), $R$ and $S$ are commutative $k$-algebras such that $R$ is noetherian, and $S$ is a finitely generated $k$-algebra, then the tensor product ...
8
votes
3answers
882 views

Kähler differentials of affine varieties

I would like to gain some intuition regarding the modules of Kähler differentials $\Omega^j_{A/k}$ of an affine algebra $A$ over a (say - algebraically closed) field $k$. Let us recall the ...
2
votes
1answer
332 views

Prime avoidance and height of an ideal

How can I use prime avoidance to prove that if $R$ is a Noetherian ring and $J\subset R$ an ideal with $\mathrm{height}(J)=n$ then there exists $x_1,\ldots,x_n\in J$ such that ...
3
votes
1answer
133 views

understanding $\mathrm{Spec}\mathbf{Z}[\mu_{p-1},\frac{1}{(p-1)p}] \cap \mathbf{Z}_p$

What's $\mathrm{Spec}\mathbf{Z}[\mu_{p-1},\frac{1}{(p-1)p}] \cap \mathbf{Z}_p$, where $\mu_{p-1}$ denotes the $(p-1)$-th roots of unity and $\mathbf{Z}_p$ the $p$-adic integers? It should be ...
1
vote
1answer
386 views

On the grade of an ideal

I need to prove the following statment (actually a special case of it). Let $R$ be a Noetherian ring, $M$ a finite $R$-module and $I$ an ideal of $R$. Then $\operatorname{grade}(I,M)\geq 2$ if ...
1
vote
3answers
579 views

A sum of noetherian modules is a noetherian module

Assume $M=N_1+N_2$ is a module, where $N_1,N_2$ are noetherian modules. How can I show that $M$ is also noetherian?
5
votes
2answers
198 views

Graded modules over $k[t,t^{-1}]$

If $R=k[t,t^{-1}]$ is a graded ring where $R_0=k$ is a field and $t\in R$ is a homogeneous element of positive degree which is transcendental over $k$, how can I prove that every graded $R$-module ...
5
votes
1answer
238 views

Projective modules over a semi-local ring

I need a little bit of help, I found that theorem, but the book doesn't prove it and gives a reference to another book that I don't have; does anyone have an idea? Let $R$ be a semi-local ring, ...
14
votes
4answers
2k views

Intuitive explanation of Nakayama's Lemma

Nakayama's lemma states that given a finitely generated $A$-module $M$, and $J(A)$ the Jacobson radical of $A$, with $I\subseteq J(A)$ some ideal, then if $IM=M$, we have $M=0$. I've read the proof, ...
33
votes
9answers
5k views

Why is the tensor product important when we already have direct and semidirect products?

Can anyone explain me as to why Tensor Products are important, and what makes Mathematician's to define them in such a manner. We already have Direct Product, Semi-direct products, so after all why do ...
3
votes
2answers
1k views

Intersection of powers of an ideal in a Noetherian ring

Given a Noetherian ring $R$ and a proper ideal $I$ of it, is it true that $\cap_n I^n=0$ as $n$ varies over all natural numbers. If not, is it true if $I$ is a maximal ideal? If not, is it true if $I$ ...
10
votes
1answer
553 views

Is a regular ring a domain

A regular local ring is a domain. Is a regular ring (a ring whose localization at every prime ideal is regular) also a domain? I am unable to find/construct a proof or a counterexample. Any help would ...
7
votes
2answers
1k views

Quotient field of a quotient ring

Given $R$ an integral domain (commutative ring with no zero divisors), and $\mathfrak P$ a prime ideal in $R$, is there a relation between the field of fractions of $R$ and the field of fractions of ...
3
votes
2answers
1k views

Notation for a polynomial ring and formal polynomials

Given that we shouldn't say that "$f(z)$ is a function", shouldn't we also not write "$p \in k[X_1, \ldots, X_n]$ is a polynomial"? Along those lines, I usually write $p(X_1, \ldots, X_n) \in k[X_1, ...
6
votes
2answers
1k views

$x$ not nilpotent implies that there is a prime ideal not containing $x$.

Let $\mathscr{N}(R)$ denote the set of all nilpotent elements in a ring $R$. I have actually done an exercise which states that if $x \in \mathscr{N}(R)$, then $x$ is contained in every prime ideal ...
12
votes
2answers
311 views

Preimaging units to units

I'm interested in (unity-preserving) homomorphisms $f: S \to T$ between (commutative, with-unity) rings $S$ and $T$ so that if $f(x)$ is a unit, then $x$ was a unit to start with. For example, an ...
4
votes
1answer
655 views

$I$-adic completion

Let $A$ be a commutative noetherian ring, and suppose that $A$ is $I$-adically complete with respect to some ideal $I\subseteq A$. Is it true that for any ideal $J\subseteq I$, the ring $A$ is also ...
5
votes
1answer
194 views

showing locally that a diagram commutes

When showing that a (natural family of) diagram of $R$-algebras for all rings $R$ commutes, why does it suffice to show that it commutes for all $R$ local with algebraically closed residue field? My ...
4
votes
1answer
771 views

Noetherian quotient rings

Let $R$ be a commutative ring and $I,J$ ideals of $R$ such that $J$ is finitely generated and the rings $R/I$ and $R/J$ are Noetherian. Are the $R$-modules $R/J$, $J/IJ$, $R/IJ$ Noetherian? Is the ...
6
votes
2answers
184 views

Description of the set of prime ideals of the $R/m^2$

Let $R$ be a commutative ring and $m\subseteq R$ be a maximal ideal. Can you describe the set of prime ideals of the $R/m^2$. Are they all maximal ?
6
votes
1answer
224 views

Modules $M$ such that the automorphism of $M \otimes M \otimes M$ induced by the permutation $(123)$ is the identity

I've been struggling with the following problem for a couple of days and I don't seem to get any further: Let $R$ be a commutative ring. I would like to get (something like) a classification of all ...
2
votes
4answers
555 views

Torsion module over PID

Suppose $p$ is irreducible and $M$ is a tosion module over a PID $R$ that can be written as a direct sum of cyclic submodules with annihilators of the form $p^{a_1} | \cdots | p^{a_s}$ and ...
11
votes
2answers
1k views

Finitely generated projective modules are locally free

Let $A$ be a commutative noetherian ring, and let $M$ be a finitely generated projective $A$-module. It is well known and easy to prove that $A$ is locally free in the sense that for every $p ...
5
votes
2answers
574 views

Completion of a Noetherian ring R at the ideal $ (a_1,\ldots,a_n)$

How can we prove that if $R$ is a commutative Noetherian ring, $\mathfrak{m} = (a_1,\ldots,a_n)$ is an ideal, then the completion of $R$ at $\mathfrak{m}$ is isomorphic to ...