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

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4
votes
4answers
181 views

Proof of the uniqueness of maximal ideal

Let $R$ be a commutative ring with $1$. Let $M$ be a maximal ideal of $R$ such that $M^2 = 0$. Prove that $M$ is the only maximal ideal of $R$.
1
vote
0answers
69 views

Relation between $H^i_I(-)$ and $H^i_J(-)$ when $I\subset J$

What is the relation between $H^i_I(-)$ and $H^i_J(-)$ (cohomological functors) when $I\subset J$ are ideals of a (local) noetherian ring?
4
votes
1answer
328 views

Irreducible ideal implies prime ideal in Dedekind Domains?

An ideal is irreducible if it can not be written as the finite intersection of strictly larger ideals. In a Noetherian ring every irreducible ideal is primary, but the converse doesn't hold. I wonder ...
4
votes
2answers
290 views

Prove that ideal generated by… Is a monomial ideal

Similar questions have come up on the last few past exam papers and I don't know how to solve it. Any help would be greatly appreciated.. Prove that the ideal of $\mathbb{Q}[X,Y]$ generated by ...
2
votes
1answer
166 views

Determinants of free modules

This should be easy, but I'm stuck. Let $A$ be a DVR and $B$ a finite algebra over $A$ that is free as an $A$-module. For $b \in B$ one can define the norm of $b$ as the determinant of ...
2
votes
1answer
94 views

$\mathrm{Hom}(R/I, R/J\otimes M)\cong ?$

Let $R$ be a Noetherian commutative ring, $I,J$ two ideal of $R$ and $M$ an $R-$module. Does anyone see the isomorphism $\mathrm{Hom}(R/I, R/J\otimes M)\cong \ldots$? Thanks.
5
votes
0answers
136 views

Question about the nullstellensatz for projective schemes

Assume that $ G $ is a graded ring. Assume that $A$ is a relevant homogeneous ideal (that is, it does not contain the irrelevant ideal $ \oplus_{n > 0}G_n$). I am having trouble proving the ...
12
votes
2answers
151 views

Uniformly solvable families of polynomials

It is a famous theorem that there is no "quintic formula", i.e. there is no formula which expresses the roots of a quintic polynomial $x^5+a_4x^4+\cdots+a_0$ in terms of $a_4,\ldots,a_0$ and rational ...
4
votes
3answers
340 views

example of a flat but not faithfully flat ring extension

I am learning commutative algebra and there is a definition about faithfully flat modules or ring extensions. I can't think of an example of a flat but not faithfully flat ring extension or module. ...
2
votes
0answers
66 views

Support of a direct sum of local cohomology modules

Let $R$ be a Noetherian ring with unit, $I$ be an ideal of $R$. Let $M$ be a finitely generated $R$ module. How can we show the following: $$\operatorname{Supp}(\bigoplus_{j\ge ...
3
votes
1answer
79 views

special case of Nagata's Lemma (Matsumura p.86)

Let $K$ be a field and $R$ a valuation ring of $K$ with maximal ideal $m_R$. Let $a \in R$ such that $1-a \in m_R$. Statement: For any $s$ that is not a multiple of the characteristic of $R/m_R$, ...
2
votes
1answer
80 views

All local cohomology modules being zero

Let $R$ be a Noetherian ring with unit, $I$ be an ideal of $R$ and let $M$ be a finitely generated $R$-module. Suppose $H_{I}^j(M)=0$ for all $j$, then how can one show that $M=IM$? The converse of ...
11
votes
2answers
464 views

Vanishing of a certain Tor

I am reading about the construction of the Affine Grassmannian in Dennis Gaitsgory's seminar notes and there are some commutative algebra facts that I am not able to figure out by myself apparently, ...
4
votes
0answers
55 views

Lattices as invertible module

Let $E$ be an etale algebra over $\mathbb{Q}$. In other words, $E$ is a finite sum of number fields. Let $L$ be a lattice in $E$, and $R$ the order associated to $L$. More explicitly, $$R=\{ e\in ...
6
votes
1answer
205 views

is the dual of a finitely generated module finitely generated?

I recently thought of this and have no idea whether over a general commutative ring the dual of a finitely generated module is finitely generated. This must be known.
2
votes
2answers
81 views

Some question on localization of polynomial ring

Let $S=A[x_1,\dots, x_r](r \geq 2)$ be a polynomial ring where $A$ is a commutative ring. Then is it true that $S=\bigcap_{i=1}^r S_{x_i}$? If $S$ is $A$-algebra and $x_i$ are not zero divisors, then ...
4
votes
1answer
71 views

Finite Projective Dimension implies non vanishing Ext

Suppose the projective dimension of a module $M$ is $n < \infty$. Does there exist a free $R$-module $F$ such that $\operatorname{Ext}^n(M, F) \not = 0$? Can't we write the free module as a direct ...
3
votes
1answer
84 views

$H_{I}^{n}(M)\cong H_{I}^{n}(R)\otimes_R M.$

Let $R$ be a Noetherian ring and $I$ an ideal of $R$. If $n$ is the cohomological dimension of $I$, then why is the following isomorphism true: $$H_{I}^{n}(M)\cong H_{I}^{n}(R)\otimes_R M.$$ The ...
6
votes
2answers
242 views

Show field of fractions is finite extension of $\mathbb{Q}$

Let $A$ be a ring which is also a finitely generated $\mathbb{Z}$-module. If $A$ is an integral domain and $K$ is its field of fractions and $K$ has characteristic zero, then why is $K$ a finite ...
3
votes
1answer
156 views

Discrete Valuation Ring and Subring of the Fractions Field

Let $R$ be a Discrete Valuation Ring, and $K$ its fractions field. Now if $B\subseteq K$ is a subring with $R\subseteq B$ then we have $$B=R \text{ or } B=K.$$ Now this seems to be a very basic ...
3
votes
1answer
289 views

proof of the Krull-Akizuki theorem (Matsumura)

This set of questions refers to the proof of the Krull-Akizuki theorem given in Matsumura's Commutative Ring Theory, pages 84-85. For those who don't have the text, i will provide the details. The ...
1
vote
1answer
122 views

Integral closure of k-algebra

Let $k$ be a field and $A$ a finitely generated algebra over $k$ that doesn't have zero divisors. Why is the integral closure of $A$ a finitely generated module over $A$ ? (edited)
9
votes
2answers
223 views

If $R$ is a commutative ring with unity and $R$ has only one maximal ideal then show that the equation $x^2=x$ has only two solutions

If $R$ is a commutative ring with unity and $R$ has only one maximal ideal then show that the equation $x^2=x$ has only two solutions. I know that $0$ and $1$ are the solutions, but I can't proceed ...
2
votes
0answers
72 views

Descent Theorem Problem

If $A$ is a finitely generated $K$-algebra which is a domain, and $p_0 \subsetneqq \cdots \subsetneqq p_t$ a chain of prime ideals, how I can show that if $F$ is the field of fractions of $A$ and ...
2
votes
0answers
76 views

Reduction of ideals in a commutative ring

Is it possible to have an infinitely generated reduction of a finitely generated ideal in a commutative ring with identity ? If yes, why ? If no, an example to this effect will be helpful. Thank ...
3
votes
1answer
63 views

Automorphism of $A[t]/(t^m)$

Let $A$ be a commutative ring and $t$ an indeterminate over $A$. If $f$ is an automorphism of the ring $A[t]/(t^m)$ satisfying $f(x)\equiv x\pmod{(t)}$ for each $x\in A[t]/(t^m)$ with $m$ a positive ...
8
votes
1answer
194 views

Artinian rings are perfect

Definition. A ring is called perfect if every flat module is projective. Is there a simple way to prove that an Artinian ring is perfect (in the commutative case)?
3
votes
1answer
135 views

Property of Hom-functor

How to prove $$\operatorname{Hom}_{R}(A,\operatorname{Hom}_{\mathbb{Z}}(R,B))\cong \operatorname{Hom}_{\mathbb{Z}}(A,B)$$ where $R$ is a commutative ring, $A$ an $R$-module and $B$ an abelian group? ...
8
votes
2answers
96 views

Why over $\mathbb{Z}/n\mathbb{Z}$ projectivity, injectivity and flatness coincide for cyclic modules?

Assume $R=\mathbb{Z}/n\mathbb{Z}$ ($n\neq0$) and let $M$ be a cyclic $R$-module. Could you tell me how to prove that $M$ is projective if and only if it is injective if and only if it is flat? And ...
8
votes
1answer
257 views

How to show that differential operator can be defined in terms of certain commutator operators

Let $U$ be any open subset of $\mathbb{R}^n$ (or, more general, of some smooth manifold). Define $\mathcal{D}_{-1}(U):=\{0\}$. For any two linear operators $A$ and $B$, the commutator operator $[A,B]$ ...
3
votes
2answers
212 views

A noetherian ring whose ideals are idempotent is artinian

I have to prove the folowing: If $R$ is a Noetherian ring, and for every ideal $I$ of $R$ we have $I = I^{2}$, then $R$ is Artinian. My first thought was to try to prove that the nilradical of ...
8
votes
1answer
521 views

An example of a commutative ring in which every primary ideal is prime

It is clear that every prime ideal in a commutative ring is primary. The converse is false; for example, in the ring $\mathbb{Z}$ the ideal $(p^2)$ is an example of a primary ideal that is not prime ...
4
votes
1answer
78 views

studying the topology of a real algebraic set

Let $f_1,\ldots,f_n \in \mathbb{R}[x_1,\ldots,x_m]$ be polynomials with real coefficients and let $I$ be the ideal that they generate. Denote by $V_{\mathbb{R}}(I)$ the corresponding real variety, ...
3
votes
2answers
404 views

radical of sum of two ideals

$I$ and $J$ are ideals in $k[x_1,\cdots,x_n]$. Show that $\sqrt{I+J}=\sqrt{\sqrt{I}+\sqrt{J}}$. I have no idea how to prove it. Can someone help?
5
votes
0answers
72 views

Flatness over Jacobson ring

I need either a reference or a counter-example to the following statement. Let $A$ be a noetherian Jacobson ring (i.e. a noetherian ring where every prime ideal $\mathfrak{p} \subset A$ is an ...
4
votes
1answer
106 views

What can be said about $p\in Spec(R)$ when $R_p$ is a field?

What can be said about $p\in Spec(R)$ when $R_p$ is a field? Especially when $R$ is local noetherian
35
votes
1answer
482 views

A ring isomorphic to its finite polynomial rings but not to its infinite one.

I was messing with the ring $k[x_1,\dots,x_n,\dots]$ of polynomials in numerable many variables in order to solve an exercise of Atiyah, and the following question came to me and made me curious: ...
6
votes
1answer
108 views

$0\to L\to R^{n}\to M \to 0$ is exact, prove $M$ is finitely presented if and only if $L$ is finitely generated.

Suppose $R$ is a ring, $0 \rightarrow L\rightarrow R^{n} \rightarrow M \rightarrow 0$ is a short exact sequence, prove $M$ is finitely presented if and only if $L$ is finitely generated.
8
votes
2answers
189 views

How does this step in the proof of the structure theorem for f.g. modules over a Dedekind domain work?

I am trying to show that every finitely generated projective module $P$ over a Dedekind domain $D$ is a direct sum of (fractional) ideals. May's notes on Dedekind domains claim the result can be ...
3
votes
0answers
82 views

why is an open faithfully-flat morphism fpqc?

Why is an open faithfully-flat morphism fpqc? In other words, why must an open faithfully flat morphism $X\rightarrow Y$ have the property that around every $x\in X$, there is an open nbhd $U$ of ...
4
votes
1answer
93 views

Monomials not in an ideal

Let $R=\mathbb{R}[x,y]$ denote the commutative ring of polynomials in two variables $x,y$ with real coefficients. Show that for each $k \in \mathbb{N}$ there exists a monomial of degree $k$ not ...
4
votes
0answers
343 views

Integral homomorphism induces a closed map on spectra

I'm trying to prove the following: Let $f:A\rightarrow B$ is a integral homomorphism (e.g. $B/f(A)$ is a integral extension). Consider $f^{*}: \operatorname{Spec}B \rightarrow ...
1
vote
1answer
67 views

Isomorphism of polynomial rings implying isomorphism of the coefficient rings [duplicate]

Let $R$ and $S$ be commutative rings. Let $x, y$ be indeterminates, and assume that one has an isomorphism $R[x] \rightarrow S[y]$ (not necessarily mapping $x$ to $y$ of course). Does this imply $R ...
7
votes
2answers
300 views

Coordinate ring in projective space. What are they?

When $X$ is an algebraic variety of affine $n$-space, then the coordinate ring of $X$ are polynomials restricted to $X$. But when $X$ is a variety of projective $n$ space, what are the elements ...
1
vote
1answer
80 views

Extension of homorphisms on a divisible R-module

Let $R$ be a principal ideal domain and let $M$ be a finitely generated $R$-module. Take $N$ a submodule of $M$ and let $P$ be a divisible $R$-module. Prove that any homomorphism $f: N \rightarrow P$ ...
2
votes
3answers
336 views

Spectrum of polynomial ring

In M. Reid's Undergraduate Commutative Algebra, the author states that if $k$ is an algebraically closed field then $\operatorname{Spec}{k[x]} = \{0\} \cup k$ (page 21). Is this correct? Instead, ...
5
votes
1answer
320 views

rational functions on projective n space

How to prove that the field of rational functions on whole of projective n space is constant functions. By rational function I mean quotients of homogeneous polynomials of same degree ...
2
votes
1answer
199 views

What does “Hauptidealsatz” mean in “Krull's Hauptidealsatz”?

What does "Hauptidealsatz" mean in "Krull's Hauptidealsatz"? Thank you very much.
1
vote
1answer
290 views

dimension of an ideal (definition)

Let $A$ be a commutative ring and $I$ an ideal. When we refer to the "dimension" of $I$, what exactly do we mean? Is it the Krull dimension of $A/I$? In particular, i am trying to understand the ...
1
vote
1answer
83 views

Annihilators of Modules

I'm stuck trying to prove that for two $R$-modules $M,N$ ($R$ commutative with a 1), then $$Ann(M+N)=Ann(M) \cap Ann (N)$$ I was trying to do double inclusion, and I can prove the RHS is contained in ...