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

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6
votes
2answers
104 views

Can you determine from the minors if the presented module is free?

Motivation (you can ignore this part): A problem in Hartshorne (II.5.8c) asks to show that if we have a coherent sheaf $\mathscr{F}$ on a reduced noetherian scheme $X$, and the function ...
1
vote
0answers
89 views

Injective hull commutes with Hom

Notation: $E_R(M)$ is the injective hull of $M$. Let $R$ be a Noetherian ring, $I$ an ideal of $R$, and $M$ an $R$-module. Then $$\mathrm{Hom}_R(R/I, E_R(M)) \cong E_{R/I}(\mathrm{Hom}_R(R/I, ...
6
votes
1answer
181 views

Formulation of Künneth theorems (definition of $\mathrm{Hom}$ and $\otimes$ of complexes)

In Rotman's An Introduction to Homological Algebra, there is written: Questions: Let $\mathbf{A}$ and $\mathbf{A'}$ be chain complexes with differentials $\partial$ and $\partial'$ respectively. ...
13
votes
2answers
393 views

Usefulness of completion in commutative algebra

After studying about the completion of a module $M$ over a ring $A$ (e.g. $I$-adic completion), I am left with the following questions: (i) What is the usefulness of the concept of completion in ...
7
votes
1answer
291 views

constructing a projection onto a variety

Consider the vector space $\mathbb{C}^n$. Given any linear subspace $S$ we can choose a complement of $T$ in $V$, i.e. $\mathbb{C}^n=S \oplus T$ and we can subsequently define a projection ...
6
votes
2answers
130 views

Is $(R_S)_{\mathfrak{p}R_S}$ isomorphic to $R_{\mathfrak{p}}$?

Let $R$ be an integral domain, let $S$ be a multiplicative subset of $R$, not intersecting $\mathfrak{p}$, where $\mathfrak{p}$ is a prime ideal of $R$. Hence $\mathfrak{p}R_S$ (the ideal generated by ...
8
votes
3answers
835 views

Commutative property of ring addition

I have a simple question answer to which would help me more deeply understand the concept of (non)commutative structures. Let's take for example (our teacher's definition of) a ring: Let $R\neq ...
2
votes
0answers
144 views

verifying if an ideal is prime

Let $R=\mathbb{Z}[\sqrt{-5}]$ and let $\mathfrak{i}=(2,1+\sqrt{-5})$ the ideal generated in $R$ by $2$ and $1+\sqrt{-5}$. I want to prove that $\mathfrak{i}$ is prime. So i considered the surjective ...
5
votes
2answers
104 views

Does $A\!\leq\!M$ and $B\!\leq\!N$ imply $A\!\otimes_R\!B\hookrightarrow M\!\otimes_R\!N$? (tensor product of modules)

Let $R$ be a commutative unital ring. What would be an example of a $R$-modules $M,N$ with submodules $A,B$, such that there does not exist an embedding of $R$-modules $$A\!\otimes_R\!B\hookrightarrow ...
3
votes
1answer
273 views

Radical/Prime/Maximal ideals under quotient maps

Let $I$ be an ideal of a ring (commutative with unity) $R$ and let $q:R\to R/I$ be the quotient map. Then there is a well known correspondence between ideals of $R$ containing $I$ and ideals of $R/I.$ ...
1
vote
3answers
110 views

Polynomial rings commute with localization

If $A \supseteq \Sigma $ is a multiplicative subset of $A$, how can I prove that there is an isomorphism of rings between $(\Sigma ^{-1} A)[X]$ and $\Sigma ^{-1} (A[X])$ ?
3
votes
2answers
284 views

Regular in codimension 1

Apologies if this is an obvious question. I've really gotten my head tangled up in knots trying to approach it from the right angle, and I'm not getting anywhere - so I thought I'd ask. A scheme is ...
4
votes
1answer
498 views

tensor product and wedge product for direct sum decomposition

If we have a real vector space $V=W_1\oplus W_2$, is it true that $W_1 \otimes W_2 = W_1 \wedge W_2 $? My guess is that this is true. The definition of the $k$-exterior power is the quotient of ...
7
votes
1answer
95 views

Can we really understand $R$ by studying $R$-modules? [duplicate]

According to Algebra: Chapter 0, the category $R\operatorname{-Mod}$ reveals a lot about $R$. However after completing the first eight chapters, I still found no examples where this happens. Can ...
6
votes
4answers
184 views

dimension of a coordinate ring

Let $I$ be an ideal of $\mathbb{C}[x,y]$ such that its zero set in $\mathbb{C}^2$ has cardinality $n$. Is it true that $\mathbb{C}[x,y]/I$ is an $n$-dimensional $\mathbb{C}$-vector space (and why)?
2
votes
2answers
80 views

Tests/ invariants for module isomorphisms

It two modules are indeed isomorphic, then it is often not too difficult to find an isomorphism since most of the time it is just the natural map. However, it takes some time for me to prove that two ...
1
vote
1answer
262 views

Prime ideals and ring extensions [duplicate]

Let $R\subset S$ be a finite extension and $P\in \text{Spec}(R)$. I'm trying to prove that$$\left \{Q\in\text{Spec}(S)\mid Q\cap R=P\right \}$$ is a finite set. Is this also true for any integral ...
3
votes
0answers
109 views

When does the ideal product commute with intersections?

Let $R$ be a commutative ring and let $\mathfrak{m} \subseteq R$ be a finitely generated ideal. If $(M_n)_{n \in \mathbb{N}}$ is a family of submodules of some $R$-module with $M_0 \supseteq M_1 ...
2
votes
1answer
54 views

Polynomial subalgebras and their fields of fractions

Let $A = k[x_1,\ldots,x_n]$, let $e_1,\ldots,e_n$ be $n$ $k$-algebraically independent elements of $A$, and $C = k[e_1,...,e_n]$. Then clearly $C \simeq A$. But suppose we have another $k$-subalgebra ...
7
votes
1answer
161 views

Finite factor ring

I'm reading the paper "How to use finite fields for problems concerning infinite fields" of Jean-Pierre Serre. In pp. 2, Serre uses the fact that, if $\Lambda\subset\mathbb C$ is a ring finitely ...
2
votes
2answers
273 views

Factorization of ideals in $\mathbb{Z}[\sqrt{5}]$

Consider the ring $R=\mathbb{Z}[\sqrt{5}]$. Let $I$ be the following ideal of $R$: $$I:=(3,1+\sqrt{5})$$ My teacher said that the following equation holds: $$I^2=(3)I,$$ but I actually can't ...
3
votes
2answers
94 views

Integral basis and Integral extensions

I have two questions: a) How can I find the integral basis of the integral closure of $\Bbb Z$ in $\Bbb Q(\sqrt{3})$.b) How can I show that an integral extension is not finite, for example how to ...
3
votes
2answers
66 views

$B \subset A$ and dim $A$ = dim $B$ means $A$ an integral extension of $B$?

I'm referring here to the Krull dimension. Is this necessarily true for commutative rings with unity? How about for finitely-generated $k$-algebras?
1
vote
1answer
229 views

calculating minimal prime ideals

Is there a "general approach" to determine the minimal prime ideals over an ideal $J$? I checked some books and didn't find a general approach. Maybe the theory of Gröbner bases is related to these ...
2
votes
1answer
77 views

Correlation between localisation & local rings

If $R$ is a commutative ring and $S$ a multiplicative subset of $R$, then one can define the localisation $S^{-1}R$ of $R$ at $S$. Now if $p$ is a prime ideal of $R$ and we set $S=R\setminus p$ then ...
3
votes
1answer
183 views

Nice proof for finite of degree one implies isomorphism?

Let $f: X \longrightarrow Y$ be a morphism of varieties over $\mathbb{C}$ and assume it is finite of degree 1, i.e. it is surjective and $$ [K(Y):K(X)] = 1 \quad \quad (*) $$ i.e. the function fields ...
2
votes
3answers
118 views

Nullstellensatz in the coordinate ring $\Gamma (X)$

One of the many statements of the Hilbert's Nullstellensatz is the following: If $k$ is an algebraic closed field, and $\mathfrak a$ is an ideal of the ring $k[T_1,\ldots,T_n]$ then $I(V(\mathfrak ...
5
votes
1answer
115 views

Artinian affine $K$-algebra

Let $K$ be a field and $A$ an affine $K$-algebra. Show that $A$ has (Krull) dimension zero (is artinian) if and only if it is finite dimensional over $K$.
1
vote
1answer
107 views

Ideals of $\mathbb{Q}(\sqrt[3]{2}) \otimes_{\mathbb{Q}}\mathbb{Q}(\sqrt[3]{2})$

I was wondering if the ring $\mathbb{Q}(\sqrt[3]{2}) \otimes_{\mathbb{Q}}\mathbb{Q}(\sqrt[3]{2})$ is a PID. I believe that it is because I think $\mathbb{Q}(\sqrt[3]{2}) ...
2
votes
1answer
94 views

Powers of ideals in a polynomial ring

Let $F$ be a field, and consider the polynomial ring $F[X_1,...,X_n]$. I am trying to prove that every power of the ideal $(X_1,...,X_k)$ is primary for $k\leq n$. For $k = n$, we have that ...
3
votes
0answers
55 views

Proof about affine varieties

Ok so I have that $k$ is algebraically closed and $F$ is an element of $k^n$, and it is a reduced polynomial. We have that $V = V(F)$. In the book it says prove that $F$ generates $I(V)$ but in my ...
1
vote
0answers
51 views

Rings noetherianos

Let $K$ be a field. Show that any subring of $K[X_{1},...,X_{n}]$ that it contains to $K$ is noetheriano. It gives an example in the one that is demonstrated not all these subrings are DFU.
5
votes
3answers
720 views

Right-adjoint functors are left-exact?

As a final exercise to VIII.1 in Algebra: Chapter 0, we are asked to prove If $\mathcal{F}\colon\operatorname{R-Mod}\to\operatorname{S-Mod}$ is a right-adjoint operator, then $\mathcal{F}$ is ...
5
votes
2answers
212 views

Is this a prime Ideal?

I wish to see wether $J=(uw -v^2, u^3 - vw, w^3 -u^5)\subset\mathbb{C}[u,v,w]$ is a prime ideal. Can somebody give me a hint to do this? Edit: More generally, I wonder wether $V(J)$, the algebraic ...
0
votes
0answers
37 views

$\mathtt{maxSpec}(\mathbb{Z}[x])=\{(p,g)\mid p\text{ is prime, }g\text{ mod }p \text{ is irred.}\}$ [duplicate]

I'm trying to prove this. This is my approach. Since $\mathbb{F}_p[x]/(\bar{g}) \cong \mathbb{Z}[x]/(p,g)$, $\mathtt{maxSpec}(\mathbb{Z}[x])\supseteq\{(p,g)\mid p\text{ is prime, }g\text{ mod }p ...
4
votes
2answers
91 views

Intersection of two localizations

Let $A$ be a commutative ring with unity. If $\mathfrak p,\mathfrak q\in \operatorname{Spec} (A)$ is it true the following equality $$A_\mathfrak p\cap A_\mathfrak q= A_{\mathfrak p\cup \mathfrak ...
4
votes
1answer
168 views

Noether Normalization Lemma for affine scheme over DVR?

Let $R$ be a DVR and $S$ a finitely generated flat $R$-algebra. How can I prove that there is a subalgebra $C$ of $S$ such that there is a finite and injective morphism $R[t_1,\dots, t_d] \rightarrow ...
2
votes
1answer
58 views

Why is the spectrum of $\mathbb{C}[X,X^{-1}]$ equal to $\mathbb{C}^*$?

Can someone help me see why the following is true?: $$\operatorname{Spec}( \mathbb{C}[X,X^{-1}])= \mathbb{C}^*$$ It was stated in something I read but I don't know why it is true. Thanks for your ...
3
votes
1answer
120 views

Finding a toric variety of a cone

I'm trying to find the toric variety associated to the cone $\sigma_0$ which is the region in the real plane with $x\geq 0$ and $y-x\geq 0.$ I found that it's dual cone is $\check{\sigma_0}$ the ...
2
votes
2answers
50 views

limits of sequences of topological rings

Let $A$ be a ring and $I$ an ideal of $A$ such that $A$ is complete in the $I$-adic topology. Let $a \in I$. Then the sequence $y_n=1-a+a^2-a^3+\cdots+(-1)^n a^n$ converges in $A$. By definition of ...
5
votes
1answer
99 views

$P/P^2$ isomorphic to $R/P$ as $R$-modules

Let $P$ be an ideal of a ring $R$. When is it true that $P^n/P^{n+1}$ are isomorphic to $R/P$ as $R$-modules for any $n$? I was trying to show that for Dedekind domains the norm of ideals is a ...
6
votes
1answer
227 views

faithfully flat ring extensions where primes extend to primes

I am interested in unital ring homomorphisms (and classes thereof) $R \rightarrow S$ of commutative rings that have the following pair of properties: $S$ is faithfully flat as an $R$-module, and ...
0
votes
2answers
64 views

How to prove $x$ doesn't lie in $R_M$

Let $R$ be an integral domain. $K$ is the field of fractions of $R$. Let $x=a/b \in K-R$ and $a \notin (b)$. How do I prove $x \notin R_M$ where $M$ is a maximal ideal containing $b$? The statement is ...
2
votes
1answer
41 views

A relation between homomorphisms from the polynomial ring zero on an ideal and homomorphisms from the quotient of the polynomial ring by this ideal

Let $n\geq 1$, $K$ be a field and $R\neq \{0\}$ a $K$-algebra. For Ideals $I$ and $J$ of $K[X_1\ldots,X_n]$ with $J\subseteq I$ consider $$ A(I)=Hom_{Kalg}(K[X_1,\ldots,X_n]/I,R) $$ and $$ ...
1
vote
1answer
343 views

Krull dimension in polynomial rings

Let $F$ be a field and $R=F[X_1,X_2,\ldots,X_n]$ be the polynomial ring in $n$ variables over $F$ and $P$ be a prime ideal in $R$, I'm trying to prove that$$\operatorname{ht}P+\dim R/P=\dim R$$where ...
4
votes
0answers
73 views

Die Relationen, welche zwischen den elementaren symmetrischen Functionen bestehen - Translation?

I am trying to find a translation of this paper either in English or French (preferably English). I am not very optimistic, but i thought of asking in case somebody is more resourceful :)
7
votes
1answer
183 views

Vandermonde identity in a ring

Let $R$ be a commutative $\mathbb{Q}$-algebra. For $r \in R$ and $n \in \mathbb{N}$ we can define the binomial coefficient $\binom{r}{n}$ as usual by $\binom{r}{0}=1$ and ...
12
votes
1answer
154 views

Original Formulation of Hilbert's 14th Problem

I have a problem seeing how the original formulation of Hilbert's 14th Problem is "the same" as the one found on wikipedia. Hopefully someone in here can help me with that. Let me quote Hilbert first: ...
1
vote
0answers
26 views

A property of linearly compact module

Let $(R,\mathfrak{m})$ be a noetherian local ring, $E$ the injective hull of $R/\mathfrak{m}$, $S=\operatorname{End}_R(E)$ and $M$ a linearly compact and discrete $R-$module. Show that ...
2
votes
1answer
34 views

Showing that the natural map into the completion is continuous

Let $M$ be an $A$-module and $M=M_0 \supset M_1 \supset \cdots$ a sequence of submodules, which we define to be a fundamental system of neighborhoods of $0$. Thus we make $M$ into a topological group. ...