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

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3
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2answers
220 views

Nilpotency of Maximal Ideal of Local Ring

What are the implications of the maximal ideal of a local ring $(A,m,k)$ being nilpotent? For example, $A$ is Artinian if and only if it is Noetherian. Any other interesting implications?
1
vote
1answer
66 views

Semicubical parabola coordinate ring

Let $C=\{(x,y)\in\mathbb{A}^2 \colon y^2-x^3=0\}$. Let $k[C]=k[x,y]/(y^2-x^3)$ be the coordinate ring of $C$. I read in a book that $k[C]$ is non a UFD. Is there a not too difficult and deep way to ...
4
votes
1answer
151 views

Exercise 4.30 of Eisenbud

I am doing exercise 4.30 from Eisenbud's Commutative Algebra With A View Towards Algebraic Geometry which I append here: Exercise 4.30: Suppose $k$ is a Noetherian ring and for every finitely ...
4
votes
1answer
115 views

$M$ is $\bigcap \operatorname{Ass}(M)$-primary

Let $R$ be noetherian ring and $M$ an $R$-module such that $\operatorname{Ass}(M)$ is a finite set. Prove that $M$ is $\mathfrak{b}$-primary, where $\mathfrak{b}=\bigcap ...
3
votes
0answers
131 views

Topology of maximal ideal space

We know that strictly positive integers with multiplication form a semigroup $S$. Let $\mathscr A=\ell ^1(S)$ with convolution. What is $\mathscr A$'s maximal ideal space? It seems enough to find ...
6
votes
1answer
133 views

Integral Closures and Affine Curves

Let $C$ be an irreducible affine curve with singular points, and let $A$ be its ring of regular functions. Since $C$ has singular points, $A$ is not integrally closed in its field of fractions, $K$. ...
3
votes
1answer
100 views

When is faithfulness transitive?

I have found the next statement in a book about separability of algebras. Let $V$ be a faithful $S$-module and $T$ an Azumaya algebra with center $S$. If $V$ is also a left $T$-module, then $V$ is ...
3
votes
1answer
163 views

Relation between localization and colimit.

I am trying to show that $S^{-1}R=\operatorname{colim}F(s)$, where $S$ is a multiplicative closed set in a commutative ring $R$ and $F$ is a functor from a filtered category $I$ to mod-$R$ and $I$ is ...
4
votes
1answer
238 views

Tensor product is zero

If $R$ is a local ring and $M$ and $N$ are finitely generated $R$-modules such that $M\otimes N=0$ then how does it follow from Nakayama's lemma that either $M=0$ or $N=0$? This is an exercise ...
2
votes
1answer
114 views

Is this square diagram cocartesian for every regular local ring?

Let $K$ be a field and $R=\{f\in K[X]\mid f(0)=f(1)\}$ the $K$-algebra obtained by pulling back $K[X]\to K\times K$, $X\mapsto (0,1)$ along the diagonal. Is the induced square \begin{eqnarray} ...
2
votes
1answer
175 views

Valuation rings

What's the spectrum of a valuation ring? How to describe morphisms from it to a scheme? Is it enough to set the image of generic point and of a maximal ideal and correspondent map of local rings?
8
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4answers
427 views

A question on definition of field of fractions

Wikipedia defines the field of fractions of a domain as The field of fractions or field of quotients of an integral domain is the "smallest" field in which it can be embedded. What does ...
4
votes
1answer
238 views

Defining multiplication on a Koszul complex

Let $R$ be a Noetherian commutative ring and $x$ and $y$ two elements in $R$. We construct the Koszul complex on $x$ and $y$. We start by the following two chain complexes: $$ C_2=0\to ...
2
votes
1answer
295 views

How can I get Macaulay2 to tell me if this ideal is prime?

I am trying to get Macaulay2 to confirm if $(y+zi,x^2 - z^2 - 1)$ is a prime ideal in $\Bbb{C}[x,y,z]$. Now as a small test, I tried to compute its radical by doing ...
2
votes
2answers
175 views

Representation of an element of the field of fractions of a Dedekind domain as a fraction of elements which are relatively prime to a given ideal

This is a generalization of this question. Let $A$ be a Dedekind domain. Let $K$ be the field of fractions of $A$. Let $I$ be a non-zero ideal of $A$. Let $\alpha$ be a non-zero element of $K$ which ...
2
votes
2answers
211 views

Product of ideals corresponding to vanishing of points is equal to their intersection

Let $k$ be some field, and let $v,v',v''$ be three distinct points in $k^3$. Let $\mathfrak{m}_v = (X_1 - v_1,X_2 - v_2,X_3 - v_3)$ be the ideal in $k[X_1,X_2,X_3]$ corresponding to the polynomials ...
2
votes
2answers
132 views

Ideal of polynomials in $k[X_1,…X_n]$ vanishing at a point $p$ is $(X_1 - p_1, …,X_n - p_n)$ [duplicate]

I'm having a little trouble following Eisenbud here: My problem is that I don't see how the isomorphism $${k[x_1,...,x_n] \over \mathfrak{m}_p} \cong k$$ is constructed. This seems a bit ...
4
votes
2answers
84 views

Question regarding two equivalent definitions of Dedekind domains

Theorem: If $A$ is a Noetherian integral domain, the following two properties are equivalent. 1) $A_{\mathfrak p}$ is a DVR for every prime ideal $\mathfrak p \neq 0$; 2) $A$ is ...
1
vote
1answer
133 views

How to form a simplicial complex?

Reference: Combinatorial Commutative Algebra by Miller and Sturmfels. I have difficulty getting started with Miller and Sturmfels, when I read the first few pages, as I do not know where simplicial ...
3
votes
2answers
92 views

Maximal ideals generate maximal submodules?

Let $\mathfrak m$ be a maximal ideal of $R$ and $M$ an $R$-module such that $\mathfrak mM\ne M$. Is it true that $\mathfrak mM$ is a maximal submodule of $M$? Thank you. (I can see this happen in ...
3
votes
1answer
181 views

What does $R_P$ mean, for a ring $R$ and an ideal $P$?

What does $R_P$ mean, for a ring $R$ and an ideal $P$? This appeared in some notes by a teacher of mine, but he didn't define this notation. He used it as follows: suppose $R$ is a commutative ring, ...
2
votes
1answer
68 views

Superfluous assumption in a counterexample to Frobenius algebras

In the wikipedia entry on Frobenius algebras, there are some examples and counter-examples. In example 5, where do you need that $\operatorname{char}(k) \neq 2$ ? I think $R:= k[x,y]/ (x,y)^2$ is ...
2
votes
2answers
113 views

Radicals of subrings

It is known that for a subring $R$ of some (commutative) ring $S$, the nilradical of $R$ $$\text{nil }R=R\cap\text{nil }S.$$ Moreover for Jacobson rings $R\subset S$, this means that the Jacobson ...
4
votes
3answers
300 views

Points and maximal ideals in polynomial rings

Let $k$ be a field, then I want to prove the following statement: for every $P=(b_1,\ldots,b_n)\in K^n$, the ideal $\mathfrak{m}_P=(x_1-b_1,\ldots,x_n-b_n)$ is maximal in the polynomial ring ...
2
votes
1answer
69 views

Arbitrary intersection of Noetherian domains is not necessarily Noetherian

If we have $R_{i}$, $i\in I$, $I$ may be infinite and each $R_{i}$ is a Noetherian integral domain with the same quotient field $K$ then it seems $R = \bigcap_{i\in I} R_{i}$ is not necessarily ...
7
votes
2answers
215 views

Tor and flat base change

There is an interesting result in Rotman's homological algebra book. Let $A$ be an $R$-module, $B$ be an $(R,S)$-bimodule and $C$ an $S$-module. (All rings commutative). Then Corollary 10.61 (in the ...
5
votes
3answers
342 views

Non-UFD integral domain such that prime is equivalent to irreducible?

In the integral domain every prime is irreducible. But the converse is not true, for example, $1+\sqrt{-3}$ is an irreducible but not a prime in ${\Bbb Z}[\sqrt{-3}]$. In a UFD, "prime" and ...
5
votes
2answers
191 views

Artinian rings and PID

Let $R$ be a commutative ring with unity. Suppose that $R$ is a principal ideal domain, and $0\ne t\in R$. I'm trying to show that $R/Rt$ is an artinian $R$-module, and so is an artinian ring if $t$ ...
4
votes
1answer
142 views

Artinian rings and associated prime ideals

Let $R$ be a commutative ring with unity. Show that $R$ is artinian ring if and only if there exists a finite length $R$-module $M$, such that $$\{r\in R \mid rm=0 ,\forall m\in M\}=(0).$$ The ...
7
votes
0answers
153 views

Divisor class group of an affine surface

In this topic the OP considers the following surface: $X=\mathcal{Z}(z^3-y(y^2-x^2)(x-1))$. (The field it's not explicitely mentioned, but for geometric reasons this can be algebraically closed.) He ...
0
votes
2answers
104 views

Projectivity of $\mathbb Q$ over $\mathbb Q\otimes_{\mathbb Z}\mathbb Q$

Consider $\mathbb Q\otimes \mathbb Q$, where $\mathbb Q$ is considered as $\mathbb Z$-algebra and consider $\mathbb Q$ as a right $\mathbb Q\otimes\mathbb Q$ module. Then is it true that $\mathbb Q$ ...
4
votes
2answers
141 views

Clarification Regarding the Tor Functor involved in a Finite Exact Sequence

Let $\cdots\rightarrow F_1 \rightarrow F_0 \rightarrow M \rightarrow 0$ be a free resolution of the $A$-module $M$. Let $N$ be an $A$-module. I saw in some notes that we have an exact sequence $0 ...
4
votes
1answer
147 views

Defining a surjective $\mathbb{Q}$-algebra homomorphism

Let $p,q$ be prime numbers. I want to define a surjective $\mathbb{Q}$-algebra homomorphism $\phi:\mathbb{Q}(\sqrt{p})[X]\rightarrow\mathbb{Q}(\sqrt{p})\otimes_{\mathbb{Q}}\mathbb{Q}(\sqrt{q})$, where ...
3
votes
1answer
126 views

An inequality of length of module

Do you know a proof for the following inequality? Suppose that $(R,m)$ is a Noetherian local ring, $q$ is an $m$-primary ideal and $M$ is a finitely generated $R$-module. Then $$ l(q^nM/q^{n+1}M) ...
13
votes
1answer
371 views

Equivalent definitions of Noetherian topological space

It is well known that we have many different definitions of noetherianity for rings. Namely, given a ring $R$, the following are equivalent: 1) every ideal of $R$ is finitely generated. 2) $R$ ...
10
votes
1answer
264 views

Completion as a functor between topological rings

In the following all rings are assumed to be commutative and unitary. Preliminaries: For any topological ring $R$ we can form its completion $\widehat{R}$ by taking all Cauchy sequences modulo null ...
6
votes
2answers
320 views

A fraction field is not finitely generated over its subdomain

I'm looking for proofs of the following fact. Suppose that $R$ is a domain which is not a field with fraction field $K$. Then $K$ is not finitely generated as an $R$-module. I know this fact is ...
3
votes
1answer
40 views

The principal fractional ideals of an integral domain form a directed partially ordered group

Let $R$ be an integral domain and $K$ be its quotient field. Let $G = \{aR: a\in K^{\times}\}$. Then $G$ is a partially ordered group under $aR\leq bR$ iff $bR\subseteq aR$. But I have hard time ...
5
votes
1answer
314 views

Tor Functor Commutes with Direct Limits

Could somebody please provide a sketch of a proof of the fact that the Tor functor commutes with direct limits? I have been trying to show that the Tor of a module with the direct limit of a ...
3
votes
1answer
82 views

Residue fields of Gorenstein local rings have finite injective dimension?

If $(R,\mathfrak m,k)$ is a Gorenstein local ring, then show that $\textrm{inj dim}_R\ k$ is finite. This was previously asked here as a second part of the question and remained unaswered, but I ...
3
votes
2answers
212 views

Projective vs Injective Resolutions

Since not all facts about projective and injective modules are not dual, i was wondering what similarities and differences are there between the information we get from projective resolutions and the ...
4
votes
2answers
437 views

Definition of a filtration on a ring, module, algebra

In most books, a graded ring/module/algebra means either a $\mathbb{N}$- or $\mathbb{Z}$-graded ring/module/algebra. But often, different gradings appear: doubly graded (spectral sequences) = ...
12
votes
2answers
354 views

The algebraic de Rham complex

Let $A$ be a commutative $R$-algebra (or more generally a morphism of ringed spaces). Then there is an "algebraic de Rham complex" of $R$-linear maps $A=\Omega^0_{A/R} \xrightarrow{d^0} \Omega^1_{A/R} ...
4
votes
2answers
136 views

$R$-linear injection [duplicate]

If $f: R^n\rightarrow R^m$ is an injective map, which is also $R$-linear, where $R$ is a commutative ring with unity. Is it true that $n$ has to be less than or equal to $m$ always?
9
votes
1answer
223 views

If $I$ is a finitely generated ideal of $A[X]$, is $I\cap A$ necessarily finitely generated for a commutative unital ring $A$?

Let $A$ be a commutative ring with $1$ and $A[X]$ the ring of polynomials in one variable over $A$. Assume $I$ is a finitely generated ideal of $A[X]$. My question is Is $I\cap A$ necessarily ...
8
votes
2answers
148 views

Principal divisors

How can i calculate the principal divisor $(f)$ where $$f = \frac{(x^{3}-1)}{(x^{4}-1)}$$ with $f\in\mathbb{F}_2(x)$. I am recently reading about the subject, so i am looking for a simple solution ...
2
votes
1answer
100 views

Morphism of graded rings

Let $\displaystyle\varphi:\bigoplus_{d=0}^\infty S_d\rightarrow \bigoplus_{d=0}^\infty T_d$ be a morphism of commutative graded rings (with identity) and suppose that there exists $d_0$ such that ...
8
votes
2answers
416 views

Inverse image of the sheaf associated to a module

In Hartshorne, Algebraic geometry it's written, that for every scheme morphism $f: Spec B \to Spec A$ and $A$-module $M$ $f^*(\tilde M) = \tilde {(M \otimes_A B)}$. And that it immediately follows ...
4
votes
1answer
127 views

Commutative ring and its group-algebra, and abelian-group-algebra as a commutative ring.

In course of discussing the algebraic structures, one of my seniors is led quite naturally to considering the $\color{red} {geometric}$ version of following: Question: Since we could consider the ...
2
votes
2answers
98 views

generators of an ideal, dimension of a vector space

Let $R$ be a local Noetherian ring (maximal ideal $m$, residue field $k$). Suppose $\{x_{1}, \ldots, x_{n}\}$ generate $m$. Is it true that dim$_{k}(m/m^2) \leq n$?