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

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3
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1answer
38 views

Injective hull of $\mathbb{ Z}_n$ [duplicate]

What is the injective hull of $\mathbb Z_n$? I know that in case $n=p$ is prime, the injective hull would be isomorphic to $\mathbb Z_{p^∞}$, but in general case, I have no idea. Can anyone be of ...
1
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2answers
35 views

Cardinality of minimal generating set of a module is constant

Let $R$ be a commutative ring with unity and $M$ be a finitely presented module over $R$. Then how to show that for any minimal generating set $S$, the cardinality is same? Edit: Thanks to Martin to ...
2
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2answers
23 views

A direct limit concerning some homomorphisms

In an algebra text there is the following argument I am stuck in the last part of which: "Let $f:B→C$ be an epimorphism in the category of $R$-modules, and $D=∑_{n=1}^∞c_nR$ be a countably generated ...
0
votes
0answers
62 views

Injective dimension and Krull dimension of a module

Let $R$ be a regular local ring and $M$ an $R$-module (not necessarily finite), then the injective dimension $\operatorname{id}_R(M)$ of $M$ is finite. When $M$ is finitely generated, we have ...
0
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1answer
24 views

A vector space in the form of a tensor product

Let $R$ be a commutative domain with fraction field $K$. It is known that $K_R$ is injective. Now, if $M_R$ is a torsion-free module and we localize at $S=R-0$ we get $M⊗_RK=S^{-1}M⊇M$. My question ...
0
votes
1answer
45 views

Isomorphism between $R$ and its dual space

Let $R$ be a finite dimensional algebra over a field $K$. If $f$ is an $R$-module monomorphism from $R$ to the dual $K$-space $\operatorname{Hom}_K(R,K)$ why it is onto? Thanks!
1
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1answer
40 views

When is $k(X)$ algebraic over $k(Y)$ for a dominant morphism $f:X\rightarrow Y$ between varieties.

Let $f:X\rightarrow Y$ be a dominant morphism between irreducible varieties over an algebraically closed field $k$. When is $k(X)$ algebraic over $k(Y)$? Is there an if and only if criterion? What if ...
3
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0answers
101 views

Ring of rational power series

Let $A$ be any commutative ring with 1. A power series $f\in A[[t]]$ is called rational if we can find a $g\in A[t]$ such that $fg\in A[t]$. It is clear that the set of rational power series forms a ...
2
votes
1answer
99 views

Help in this notation in Fulton's Algebraic Curves book

I'm reading Fulton's Algebraic Curves book, I'm stuck in the following proposition (page 105): In fact, what I didn't understand is the following notation in the proof of this proposition: Why ...
0
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1answer
61 views

Proof that presheaf is a sheaf for Spec

Atiyah Macdonald define presheaf (chapter 3, exercise 23) on the base of $Spec(A)$, where $A$ is commutative ring with $1$, as follows $$ \mathfrak{F}(X_f) = A_f, $$ where $X_f$ is a basic open set ...
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0answers
54 views

Direct image of the exceptional divisor along a blow-up

Let $X=\mathrm{Spec}(k[x_1,\ldots,x_n])$ for $n\geq 2$, and let $\mathcal{I}=\widetilde{I}\subseteq\mathcal{O}_X$ for an ideal $I\subseteq k[x_1,\ldots,x_n]$. Let ...
1
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2answers
149 views

Every element in a ring with finitely many ideals is either a unit or a zero divisor.

I came across the above proposition on mathstackexchange If every nonzero element of $R$ is either a unit or a zero divisor then $R$ contains only finitely many ideals. the link asks a different ...
7
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3answers
209 views

In $\mathbb{Z}/(n)$, does $(a) = (b)$ imply that $a$ and $b$ are associates?

[Update: Based on the hints provided by @zcn and @whacka, I believe I have found a solution. See my answer below.] Below, $R$ is a commutative ring with $1$. In John J. Watkins' Topics in ...
0
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2answers
49 views

Preimages of coprime ideals

Assume $R,S$ are commutative rings, $f:R\to S$ is a surjective ring homomorphism and $I,J$ are coprime ideals in $S$. Must $f^{-1}(I)$ and $f^{-1}(J)$ be coprime in $R$?
3
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1answer
117 views

Endomorphisms of the maximal ideal of a local ring

Let $R$ be a commutative local ring with maximal ideal $\mathfrak{m}$. Is it true in general that $\text{Hom}_R(\mathfrak{m},\mathfrak{m})\cong \text{Hom}_R(\mathfrak{m}, R)$? What if the Krull ...
1
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0answers
53 views

Quotient of local ring is of finite length

My objective is to show that $\mathcal{O}_{P}/(f,g)$ is of finite length as a $\mathcal{O}_{P}$-module. $\mathcal{O}_{P}$ is the local ring of $P = (0, 0)$. In other words it's $k[x, ...
1
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1answer
80 views

The spectrum of a commutative ring with unity and its “topology”

Let $\operatorname{Spec}(R)$ be the set of prime ideals in the commutative ring with unity $R$, and let $\mathfrak a$ be some ideal. Show that we get a topological space if we define the closed sets ...
0
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1answer
60 views

Completion of quotient of polynomial ring

Hartshorne's Algebraic Geometry uses the following facts on page 35 without proof: The completion of $(k[x,y]/(y^2-x^2-x^3))_{(x,y)}$ is $k[[x,y]]/(y^2-x^2-x^3)$ and that of ...
2
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1answer
49 views

Von Neumann regular but not self-injective ring

I want an example of a von Neumann regular ring which is not self-injective. My thanks go to anybody answering.
1
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1answer
29 views

Subbase of a topology containing prime ideals (commutative ring)

Let $A$ be a commutative ring. Prove that the set of the ideal primes of $A$, along with $A$, is a subbase of some topology on (the subjacent set of) $A$ and that the complements of the prime ideals ...
2
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1answer
41 views

A statement related to Hilbert Nullstelensatz

By Hilbert Nullstelensatz we know that for any field $k$, every maximal ideal of $k[x_1, ..., x_n]$ has residue field a finite extension of $k$. I also did an exercise which goes: any integral domain ...
2
votes
2answers
72 views

Minimal injective resolution of a module

Let $R$ be a commutative Noetherian ring and $M$ an $R$-module. Let $0\rightarrow M \rightarrow E^{\bullet}$ be a minimal injective resolution of $M$ and $0\rightarrow M\rightarrow I^{\bullet}$ be an ...
0
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1answer
28 views

Does extension of scalars take Noetherian modules to Noetherian modules?

Suppose $A$ is a commutative ring with unity, and $B$ is an $A$-algebra. If $M$ is a Noetherian $A$-module, is $M \otimes_A B$ Noetherian as a $B$-module? Note that there are no finiteness conditions ...
7
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2answers
195 views

Showing that $x^3+y^3+z^3=0$ is not rational

Is there a short proof that $F:x^3+y^3+z^3=0$ in $\mathbf{P}^2$ is not rational, apart from using the genus? Perhaps this is an elliptic curve, so every morphism $\mathbf{P}^n\rightarrow F$ is ...
4
votes
1answer
147 views

$k$-algebra homomorphism of the polynomial ring $k[x_1,\dots,x_n]$

Let $\phi:k[x_1,\dots,x_n]\mapsto k[x_1,\dots,x_n]$ be a $k$-algebra homomorphism with $\phi(x_i)=f_i$, where $k$ is algebraically closed and has characteristic zero. I have the following questions: ...
2
votes
1answer
35 views

Example of a ring which is not CM at all its prime ideals

A commutative ring $A$ is said to be CM at a maximal ideal $\mathfrak{m}$ if and only if $Depth(A_{\mathfrak{m}})=Krull(A_{\mathfrak{m}})$. What is an example of a connected commutative ring $A$ which ...
4
votes
1answer
85 views

Flatness after dividing out a minimal prime ideal

Let $A \hookrightarrow B$ be an extension of finitely generated, reduced $k$-algebras, where $k$ is a field of characteristic zero such that $B$ is a free $A$-module of finite rank. Let $A$ be an ...
0
votes
1answer
50 views

A nonregular local ring [duplicate]

Consider the ring of the formal power series $k[[T_1,\ldots,T_n]]$ ($k$ algebraically closed) where $\mathfrak m$ is the maximal ideal. If $f\in\mathfrak m^2$, why $$\frac{k[[T_1,\ldots,T_n]]}{(f)}$$ ...
1
vote
1answer
35 views

Height one prime avoidance in normal domains [duplicate]

Let $R$ be a Noetherian normal domain. Let $X$ be the set of height one prime ideals of $R$, and let $\mathfrak p \in X$. Can one have $$ \mathfrak p \subseteq \bigcup_{\mathfrak q \in X \setminus ...
0
votes
1answer
76 views

Some residue field

Consider a prime ideal $\mathfrak{p}\in\mathrm{Spec} \ \mathbf{Z}[x]$; the residue field at $\mathfrak{p}$ is the fraction field of $\mathbf{Z}[x]/\mathfrak{p}$. Can we classify the residue fields? I ...
1
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1answer
72 views

Are $\mathbb{Q}$ or $\mathbb{Z}$ flat modules?

I have the following three question about flat modules. Why is not $\mathbb{Z}$ a flat $\mathbb{Z}$-module. Why is $\mathbb{Q}$ a flat module $\mathbb{Z}$-module. I need an example of a module which ...
1
vote
1answer
29 views

Torsion-free quotient of integer polynomial ring

Consider the ring of polynomials $\mathbb{Z}[x,y]$ and let $I$ be the ideal $(xy,x+y)$. Is the quotient $\mathbb{Z}[x,y]/I$ torsion-free as a $\mathbb{Z}$-module? How does one approach this type of ...
1
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1answer
42 views

Another description of injective hull

Let $I$ be an injective module containing a module $M$, let $M_1$ be a submodule of $I$ maximal with respect to the property that $M_1∩M=0$, and let $M_2$ be a submodule of $I$ containing $M$ maximal ...
1
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0answers
58 views

Deduction of usual Cayley-Hamilton Theorem from “Determinant Trick”

Here is a statement of a standard theorem in commutative algebra (see page 60 of this book): Theorem. ("Determinant Trick") Suppose that $R$ is a commutative ring with $1$. Let $M$ be a finitely ...
2
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0answers
77 views

If $R$ is a domain and $M$ a finitely generated $R$-module, is it true that $\bigcap_{f\in M^{*}}\ker{f}=\operatorname{Tor}M$?

Let $R$ be a domain and $M$ a finitely generated $R$-module. Let $M^{*}=\hom_{R}(M,R)$. Let Tor$M$ be the torsion submodule of $M$. It it true that $$\displaystyle\bigcap_{f\in ...
3
votes
2answers
91 views

On Hilbert's Nullstellensatz Theorem

I was reading Ravi Vakil's notes on his website and he states the Hilbert Nullstellensatz (3.2.5.): If $k$ is any field, every maximal ideal of $k[x_1, ..., x_n]$ has residue field a finite extension ...
1
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2answers
129 views

Nonintegral element and a homomorphism

Assume $R\subseteq S$ are rings. Choose $x\in S$ nonintegral over $R$. I want to define a homomorphism from $R[x^{-1}]$ to a field which maps $x^{-1}$ to zero. I was trying to show that ...
2
votes
1answer
48 views

Maximal homogeneous ideals of a graded $k$-algebra.

Let $k$ be an algebraically closed field, and let $A$ be a finitely generated commutative $k$-algebra. Given any maximal ideal $\mathfrak{m}\subset A$, we can form the quotient to obtain a map $A\to ...
0
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2answers
65 views

Cohen-Macaulay and regularity

I know this is a simple question but to make sure....: $A$ is a commutative ring which is Cohen-Macaulay and for every maximal ideal $\mathfrak{m}$ in $A$ if $\dim A_{\mathfrak{m}}=\dim A$ then ...
3
votes
2answers
42 views

Kahler differentials and quotient rings.

I am dealing with some nice rings that are always isomorphic to some fairly nice quotient ring of a polynomial ring. A typical example is: $$ \mathbb{C}[X,XY,XY^2] \cong ...
1
vote
1answer
121 views

Writing $I= (xz-y^2, yt- z^2)$ as an intersection of prime ideals

I need to write the ideal $I= (xz-y^2, yt- z^2) \subset R = \mathbb{K}[x,y,z,t]$ as intersection of prime ideals. Any idea? For the moment, I've noticed that $I$ is radical, then it suffices to ...
0
votes
2answers
90 views

defining the group law on elliptic curves in general

Let $k$ be an arbitrary field and $C \subset \mathbb{P}^2(k)$ an elliptic curve. In order to define the group law on $C$ we need to establish some geometric facts first, e.g. Any line intersects $C$ ...
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0answers
35 views

A question regarding a lemma in Perrin's Algebraic Geometry.

Algebraic Geometry by Perrin says the following: Let $k$ be an uncountable algebraically closed field and let $K$ be an extension of $k$ whose dimension is at most countable. Then $K=k$. He ...
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0answers
52 views

Krull dimension of localization

If $R$ is a commutative ring and $m$ a maximal ideal therein, then what are the conditions for the Krull dimension of $R$ equaling to the Krull dimension of $R_m$?
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1answer
34 views

Definition of degree of commutative ring $d(A) $ based on Hilbert polynomial

I'm studying chapter 11 (Dimension Theory) in Atiyah / Macdonald - Intro to Commutative Algebra. Let $ A $ be a Noetherian local ring with $\mathfrak{m}$-primary ideal $\mathfrak{q}$. The book defines ...
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0answers
69 views

Going Down Theorem, AM

I'm trying to understand the proof of the going down theorem in Introduction to Commutative Algebra by Atiyah and Macdonald. My main confusion is when they say it suffices to show that $B_{\mathfrak ...
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vote
3answers
118 views

Some practical questions on cohomology and the ring $\mathbf{Z}[x]/(x^2)$

So I know that the cohomology ring of $S^n$ is $\mathbf{Z}[x]/(x^2)$ with "$x$ in degree $n$"; if we ignore the grading then this ring fails to distinguish the spheres. What is actually meant by "in ...
3
votes
0answers
83 views

Tensor products over monoids : Element structure

Let $A$ be a (commutative) monoid. Let $M$ be a right $A$-set and let $N$ be a left $A$-set. Then we can construct the tensor product $M \otimes_A N$, which is a set (of even $A$-set when $A$ is ...
0
votes
1answer
29 views

Intuition behind the abstract definition of a node (singularity)

Look at the following definition of an ordinary double point (node). The source is the book:"Freitag, Kieh - Etale Cohomology and the Weil Conjectures:" I don't understand the geometry behind ...
0
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1answer
72 views

Canonical ring map

Let $\chi:\mathbf{Z}\rightarrow A$ be the canonical map to a ring $A$, and let $p$ be a prime ideal of $A$. Then I claim that $\chi^{-1}(p)=(\mathrm{char} \ k(p))$ where $k(p)$ is the residue field at ...