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

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1answer
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Question on Modules [closed]

Find an example of a module $M$ over a ring $A$ such that $M\neq0$ but $M\otimes(A/\mathfrak{m})=0$ for every maximal ideal $\mathfrak{m}$ of $A$, where the tensor product is taken over $A$.
4
votes
1answer
219 views

Example of rings of the same positive characteristic that do not embed into their tensor product?

I'm overcoming my fear of tensor products, and the following exercise got me wondering: Give an example of commutative rings $A$ and $B$ with $\operatorname{char}A=\operatorname{char}B$ such that ...
2
votes
0answers
36 views

Does $M$ finitely presented and $N$ finitely generated imply Hom$_R(M,N)$ f.g. when $R$ is not Noetherian? [duplicate]

If $R$ is a non-Noetherian ring, $M$ is a finitely presented $R$-module, and $N$ is a finitely generated $R$-module, does it hold that Hom$_R(M,N)$ is a finitely generated $R$-module? We tried ...
0
votes
2answers
86 views

What are the closed subsets of $\operatorname{Spec}(\mathbb{Z})$?

I'm trying to find what the closed subsets of $\operatorname{Spec}(\mathbb{Z})$ are. I know that the prime ideals of $\mathbb Z$ are the ideals generated by prime numbers, i.e., the prime ideals of ...
1
vote
2answers
101 views

Can we find a subset of $Spec(R)$ not quasi-compact?

If $R$ is a commutative ring with unit, we can easy prove that $Spec(R)$ is quasi-compact. However can you give me an example of $R$ such that a subset $A \subset Spec(R)$ isn't quasi-compact?
1
vote
1answer
126 views

proving that $\dim A[X] = \dim A + 1$ (Matsumura)

Let $A$ be a Noetherian ring and $X$ an indeterminate over $A$. I am having trouble understanding Matsumura's proof (Commutative Ring Theory, Theorem 15.4) that $\dim A[X] = \dim A + 1$. Below, i ...
0
votes
1answer
47 views

A completely reducible module is isomorphic to its associated graded module?

If $F.(M)$ is a (finite) filtration of a finitely generated module $M$ that is completely reducible, then $M \cong \operatorname{gr}_{F.(M)}$? Let $0=F_{n+1}(M) \leq F_{n}(M) \leq \cdots \leq ...
3
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0answers
78 views

Module of smooth vector fields

I want to show that the module of smooth vector fields is a free module over the ring of infinitely differentiable functions on some open subset of Euclidean space. I understand how to prove this from ...
1
vote
1answer
131 views

A question on the Chinese Remainder Theorem

This is a question from Lang's ANT, Thm 2 (ch.7, $\S2$). Let $k$ be a number field and $A$ its adele group. In the proof, Lang states Given $x\in A$, let $m$ be a rational integer such that ...
2
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0answers
108 views

Irreducible polynomials as formal power series

I'm studing the ring of formal series with complex coefficients $\mathbb{C}[[x]]$. I proved that the polynomial $y^2-x^3-x^2$ is irreducible in $\mathbb{C}[x,y]$ but reducible in $\mathbb{C}[[x,y]]$. ...
1
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1answer
111 views

Global dimension.

What is the global dimension of $\mathbb{Z}_{(p)}$ and $\mathbb{Z}_{(p)}/t\mathbb{Z}_{(p)}$, where $\mathbb{Z}_{(p)}$ is the local ring, $p$ prime and $p \mid t$? What is the global dimension of ...
2
votes
1answer
147 views

augmented algebras and their morphisms

Let $R$ be a commutative unital ring and $A$ an associative (unital) $R$-algebra. What is an augmented $R$-algebra? A (unital) $R$-algebra $A$, together with a (unital) ring morphism $\varepsilon: ...
4
votes
1answer
359 views

Does Localization Commute with Direct/Inverse Limits

Let $A$ be a ring and let $M_n$ be $A$-modules. For a prime ideal $P$ in $A$ is it true that $$(\varprojlim_n M_n)_P=\varprojlim_n (M_n)_P\text{ and } (\varinjlim_n M_n)_P=\varinjlim_n (M_n)_P?$$ If ...
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1answer
68 views

Given a f.g. module with this property [closed]

Give a finitely generated $R-$module with at least one submodule which has a infinite generator set ($R$ need not to be Noetherian).
7
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1answer
144 views

Derived category of certain ring

I'm interested in the structure of $D^b(R)$, where $R=k[x]/(x^n)$. How one can describe this category? What is the list of indecomposable objects in this category?
6
votes
1answer
144 views

Determinant vanishing over polynomial ring

Let $R=\mathbb C[t_1,\ldots,t_N]$ be a polynomial ring in some number of variables. Assume that $f_{ij}\in R$ are homogeneous linear polynomials for $1\le i,j\le n$. If $\det(f_{ij})=0$, I can ...
1
vote
1answer
52 views

A set of prime factors of an integer in $\mathcal{O}_k$

I've got a basic question from Thm 2 (ch.7, $\S2$) of Lang's Algebraic Number Theory. Let $k$ be a number field and $A$ its adele group. Let $S_{\infty}$ be the set of Archimedean absolute values of ...
6
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2answers
116 views

Concept of a subring in Atiyah-Macdonald's book

I think this definition is wrong, because nothing guarantees that the subring is closed to additive inverses. Thanks
1
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1answer
134 views

If for all $r\in R$ the element $ar+1$ is invertible in $R$, then $a$ belongs to the Jacobson radical

Let $R$ be a commutative ring with unity, and let $a$ be a fixed element of $ R $. Suppose that for every $ r \in R $, $ ar + 1 $ is invertible in $ R $. Show that $ a $ belongs to the Jacobson ...
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0answers
57 views

Are there in $(\mathbb{C}[x,y,z]/(x^3+y^3+z^3))_{x}$ exactly $12$ lines?

Let $R=\mathbb{C}[x,y,z]/(x^3+y^3+z^3)$ be the coordinate ring of the affine variety defined by the equation $x^3+y^3+z^3=0$. We can consider the localization in the element $x$, denoted by $R_x$. I ...
3
votes
2answers
104 views

(Integer) Variant of Hilbert’s irreducibility theorem

Let $P\in{\mathbb Q}[X,Y]$ such that $P(x,.)$ has an integer root for any integer $x\in{\mathbb Z}$. Does it follow that $P$ has factors of the form $Y-Q(X)$ for some $Q\in{\mathbb Q}[X]$, and does ...
2
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0answers
71 views

How to prove that a DVR is not complete

My question is inspired by a comment in this topic. How to prove that $R=\mathbb C[x]_{(x)}$ is not complete in the topology of its maximal ideal? One knows that $R$ is a DVR, and its field of ...
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1answer
453 views

Does quotient commute with localization?

Let $R$ be a commutative ring, and $I \subset R$ an ideal. If we choose an element $x \in R$ we can consider $(R/I)_x$ and $R_x/I_x$. In general, does localization commute with quotient? i.e. $(R/I)_x ...
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0answers
31 views

Submodules of a Noetherian module are finite intersections of irreducible submodules [duplicate]

If $M$ is a Noetherian $R$-module then every submodule of $M$ is a finite intersection of irreducible submodules. Please show me the way how to get the proof of this statement.
6
votes
3answers
372 views

Is $\mathbb{C}[[x]] \simeq \mathbb{C}[x]_{(x)}$?

Let $\mathbb{C}[x]$ be the ring of polynomials and $\mathbb{C}[[x]]$ the formal power series. Is $\mathbb{C}[[x]] \simeq \mathbb{C}[x]_{(x)}$? Is it true? Is there a geometric interpretation of ...
0
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1answer
53 views

Knots and reducible spectra $\mathbb{C}[\![x,y]\!]/I$

Let $I=(y^2-x^3-x^2)$ be an ideal of $\mathbb{C}[x,y]$. I don't know why $\operatorname{Spec}(\mathbb{C}[x,y]/I)$ is irreducible but $\operatorname{Spec}(\mathbb{C}[\![x,y]\!]/I)$ is reducible. Do you ...
0
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1answer
116 views

$\operatorname{Spec}(R)$ not connected iff $R$ is a product of two rings [duplicate]

Let $R$ be a commutative ring. How can I prove that $\operatorname{Spec}(R)$ is not connected if and only if $R$ is isomorphic to the product of nonzero ring $A_1$ and $A_2$? If we consider ...
6
votes
1answer
275 views

Prove that $k[x,y,z,w]/(xy-zw)$, the coordinate ring of $V(xy-zw) \subset \mathbb{A}^4$, is not a unique factorization domain

I want to show that $k[x,y,z,w]/(xy-zw)$, the coordinate ring of $V(xy-zw)\subset\mathbb{A}^4$, is not a unique factorization domain. Morally, all we need to do is find some nonzero element that ...
3
votes
1answer
323 views

Working out the normalization of $\mathbb C[X,Y]/(X^2-Y^3)$

I'm trying to identify the normalization of the ring $A := \mathbb C[X,Y]/\langle X^2-Y^3 \rangle$ with something more concrete. First, $X^2-Y^3$ is irreducible in $\mathbb C[X,Y]$, making ...
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vote
1answer
84 views

$\operatorname{Spec}(\mathbb{C}[[x,y]])$.

Let $\mathbb{C}[[x,y]]$ be the ring of the formal series with coefficients in $\mathbb{C}$. I have to find $\operatorname{Spec}(\mathbb{C}[[x,y]])$. I think that it is a local ring because it ...
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0answers
169 views

Find prime ideals

I have two questions in order to find general methods to find prime ideal: the first one is from Ravi Vakil: 1) Suppose $I=(wz-xy,wy-x^2,xz-y^2) \subset k[x,y,z,w]$. How can I prove that ...
3
votes
2answers
145 views

Is $(x^2+y^2-1,z-iy)$ a prime ideal in $\mathbb C[x,y,z]$?

Is $(x^2+y^2-1,z-iy)$ a prime ideal in $\mathbb C[x,y,z]$? How can I prove it? I need this to decompose the algebraic set $V(x^2+y^2-1,x^2-z^2-1)$ into irreductible components.
6
votes
2answers
141 views

Can an integral domain be embedded in a proper quotient of itself?

Does there exist an integral domain $R$ which has a proper ideal $J$ so that there exists an injective ring homomorphism $\phi \colon R \to R/J$? If yes, what are suitable assumptions on $R$ to ...
2
votes
1answer
54 views

Are automorphisms of extensions trivial?

Here is a statement for abelian categories which seems so basic I'm feeling embarrassed to have to ask whether it's true in general. Suppose $0 \to A \to B \to C \to 0$ is an exact sequence with ...
0
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2answers
97 views

Localizations of a quotient ring

Let $R=\mathbb{C}[x,y,z]/(x^3+y^3+z^3)$ and $f=x^2$. I have to know what is $R_f$ (localization in the element $x^2$). How can I describe $\operatorname{Spec}(R_f) $ and ...
0
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3answers
91 views

Is $\operatorname{Spec}(\mathbb{C}[x,y]/(y-x^2))$ the same as $\operatorname{Spec}(\mathbb{C}[x])$ ?

Let $R=\mathbb{C}[x,y]/(y-x^2)$. We have that $$\operatorname{Spec}(R)=\{(0),(x-a,y-a^2),(y-x^2)\}. $$ But if we consider the quotient ring $\mathbb{C}[x,y]/(y-x^2) \simeq \mathbb{C}[x]$. But ...
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2answers
98 views

Show that the ring of holomorphic functions on the unit disc is not a local ring

I'm asked to show that the ring of holomorphic functions on the unit disc $\{z \in \mathbb{C} \mid |z| < 1\}$ is not a local ring. I'm quite sure that this is not a difficult proof, and I've ...
2
votes
1answer
69 views

Finiteness of a field that is a homomorphic image of a polynomial ring

Let $S=\mathbb F_q[x]$ be the polynomial ring over the finite field $\mathbb F_q$. If $I=\langle p(x)\rangle$ is a maximal ideal of $S$ ($p(x)$ is irreducible), then the field $S/I$ is also a finite ...
2
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0answers
75 views

Completion of intersection of prime ideals

Let $R$ denote the ring of convergent power series over $\mathbb{C}$ in $n$ variables (which is a Noetherian, excellent, local ring). For any finite set of prime ideals one has that the ...
3
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1answer
49 views

Relationship between maps and maps of rings

I take this exercise from Ravi Vakil's book. Consider the map of complex manifolds sending $\mathbb{C} \rightarrow \mathbb{C}$ via $x \mapsto y=x^2$. We interpret $\mathbb{C}$ as the $x$-line and ...
3
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1answer
68 views

$(\mathbb{C}[x,y]/(xy))_x \simeq \mathbb{C}[x]_x$

I have to prove that there is an isomorphism $(\mathbb{C}[x,y]/(xy))_x \simeq \mathbb{C}[x]_x$. Geometrically the situation is clear because we have that $Spec(S^{-1}A)$ is the set of prime ...
6
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1answer
205 views

When is a field a nontrivial field of fractions?

If we take any integral domain, then we can define a field of fractions by taking equivalence classes of ordered pairs of elements, the same way that the rational numbers are constructed from the ...
6
votes
2answers
256 views

characterization of projective/injective/flat modules via $\operatorname{Hom}$ and $\otimes$

Let $R$ be a commutative unital ring and $M$ an $R$-module. Then $M$ is projective iff $\operatorname{Hom}(M,-)$ is exact, injective iff $\operatorname{Hom}(-,M)$ is exact, and flat iff $M\otimes-$ is ...
3
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1answer
65 views

$\mathrm{Ann}_RA+\mathrm{Ann}_RB\,\subseteq\mathrm{Ann}_R\,\mathrm{Ext}^n_R\!(A,B)$?

Let $R$ be a commutative unital ring and $r\!\in\!R$. Let $A$ and $B$ be $R$-modules. Does $rA\!=\!0$ or $rB\!=\!0$ imply $r\mathrm{Ext}^n_R(A,B)=0$ for all $n\in\mathbb{N}$? For $n=0$ it holds, but ...
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1answer
180 views

Two questions on Hilbert's Nullstellensatz.

We have this strong version of Hilbert's Nullstellensatz: If $k$ is any field, every maximal ideal of $k[x_1, \dots, x_n]$ has residue field a finite extension of $k$. Now I have two questions: ...
3
votes
2answers
164 views

$\operatorname{Spec}(k[x])$ has infinite points.

Let $k$ be a field. I have to prove that $\operatorname{Spec}(k[x])$ has infinite points. If $k$ is infinite it is obvious: in fact there are infinite maximal ideals $(x-\alpha_i)$, with $a_i ...
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0answers
68 views

How can I calculate $\mathrm{Spec}(\mathbb{Z}_{(3)})$? And $\mathrm{Spec}(\mathbb{Z}_3)$?

Let $\mathbb{Z}_{(3)}$ be the localization (in $\mathbb{Z}$) of the ideal generated by $3$. So I have to put in $\mathbb{Z}$ all the inverses of the complement of $(3)$. How can I calculate ...
2
votes
1answer
91 views

Intersecting kernels of localization homomorphisms (Atiyah-MacDonald Ch. 4 problem 10d)

In Atiyah-MacDonald Ch. 4 Problem 10, we have an arbitrary ring $A$ and various prime ideals $\mathfrak{p}$ and we are considering the kernels of the localization maps $A\rightarrow A_\mathfrak{p}$, ...
1
vote
2answers
124 views

Compute the Zariski-closure of two sets

Compute the Zariski-closure of the following two sets: $X = \{(z_1, z_2) \in \mathbb{C}^2 \mid |z_1|^2 + |z_2|^2 = 1\}$ in $\mathbb{C}^2$. $X = \{z \in \mathbb{C} \mid z = n \in \mathbb{N}\}$ in ...
6
votes
1answer
150 views

Elementary equivalence of polynomial rings

In his notes on the model theory of valued fields, Lou van den Dries mentions in bypassing that the polynomial ring over the complex numbers $\mathbb{C}$ is not elementarily equivalent to the ...