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

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The number of generators of the ideal of an algebraic set

Suppose $X\subset k^n$ is an $m>0$ dimensional quasi-affine variety. How can I choose $m$ polinomials generating the ideal $I$ of $X$?
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89 views

Kernel and direct sum

Let $R=k[x_1,\ldots,x_7]$ be a polynomial ring over field $k$ and $I=\bigcap_{i=1}^4 \mathfrak{p}_i$ where $\mathfrak{p}_1=(x_1,x_3,x_5,x_6), \mathfrak{p}_2=(x_1,x_3,x_4,x_6), ...
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2answers
84 views

Computing kernel

Let $I,J$ be two ideals of Noetherian ring $R$. How to compute kernel of following homomorphism directly: $$\phi: R/I\oplus R/J\to R/(I+J) $$ $$(a+I,b+J)\to (a-b)+I+J $$
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1answer
119 views

Brute Force Algebraic Geometry: interpretation of algebraic extensions of field of rational functions

I am trying to affirm the following (without use of dimension theory): Let $V_{1},V_{2}$ be the zero-loci of prime ideals $I_{1},I_{2}<\mathbb{K}[x_{1},\ldots,x_{d}]=:R$ such that ...
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1answer
219 views

If $A$ is reduced, Spec $A$ has no embedded points

I've partly solved the following exercise of Vakil's FOAG, but I am not sure I got the last part right. Could some take a look? 5.5.C. EXERCISE (ASSUMING (A)). Show that if $A$ is reduced, Spec ...
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126 views

If for $A$ a commutative nonzero ring $A^m ≅ A^n$ as $A$-modules, then $m = n$

This is the problem I need to solve: Let $A$ be a nonzero ring. Show that if $A^m ≅ A^n$, then $m = n$. The book I got this problem from suggests using the following method to solve it: Let ...
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How to prove the ring $R_J$ is Noetherian without use the theorem of I. S. Cohen?

First define $R:=k\left[{\{X_i\}}_{i\in\mathbb{N}}\right]$ where $k$ is a field (it could be an integral domain as $\mathbb{Z}$ too for example). This ring is an integral domain and it is not ...
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123 views

The kernel of homomorphism of a local ring into a field is its maximal ideal?

I have a question about the proof of Theorem 3.2. of Algebra by Serge Lang. In the theorem $A$ is a subring of a field $K$ and $\phi:A \rightarrow L$ is a homomorphism of $A$ into an algebraically ...
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1answer
82 views

Homomorphism between finitely generated $k$-algebras of (Krull) dimension 1 preserving maximal properties of ideals.

Let $k$ be any field (may not be algebraically closed), $A$ and $B$ are two finitely-generated $k$-algebras of (Krull) dimension 1. Suppose $f : A \rightarrow B $ be a $k$-algebra homomorphism. I want ...
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86 views

Separability of field extensions

I'm trying to figure out if these statements are equivalent for an arbitrary field extension $L/k$, such that $\text{char} (k) = p$ and $A$ is a reduced commutative $k$-algebra. $1)$ $L/k$ is ...
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62 views

Example 4.3.19 in Liu: unramification with schemes and numbers

In exemple 4.3.19 of Liu's book one hase $L/K$ an extension of number fields with integer rings $\mathcal{O}_L$ and $\mathcal{O}_K$, $\mathfrak{q}\subseteq\mathcal{O}_L$ a prime ideal and ...
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51 views

Is it true that $\operatorname{colim}\operatorname{Hom}_{R_i}(M,N)$ is isomorphic to $\operatorname{Hom}_{\lim R_i}(M,N)$?

Suppose we have a system of ring homomorphisms $\cdots \to R_{i+1} \stackrel{f_i}{\to} R_i \to \cdots \stackrel{f_0}{\to} R_0$ (we may assume all the maps are injective, but that's not necessary for ...
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1answer
75 views

How to distinguish vector space from $k$-algebra?

Let $A$ be a finitely generated $k$-algebra with no nilpotents. What do I need to show in order to prove it's a finite dimensional vector space over $k$? For example, is it enough to show that ...
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1answer
86 views

$(\mathfrak{a}+\mathfrak{b}:f)\stackrel{?}{=}(\mathfrak{a}:f)+(\mathfrak{b}:f)$

Suppose $R$ is a commutative ring with unit and $\mathfrak{a},\mathfrak{b}$ are ideals of $R$. We can define $(\mathfrak{a}:f)=\{g\in R\mid gf\in \mathfrak{a}\}$ for $f\in R$. I can see that ...
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1answer
48 views

Proof of 4.3.12 in Liu: dimension of fiber and flatness

I don't understand a detail in the proof of theorem 4.3.12 in Liu: we have a flat morphism $f:X\to Y$ of locally Noetherian schemes, $x\in X$ and $y=f(x)$. We want to proove that ...
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2answers
177 views

Show that $Ker(f)$ is finitely generated, when $f: M \rightarrow A^n$ is a surjective A-module homomorphism and M is finitely generated

This is the problem I am attempting: Let $M$ be a finitely generated A-module and $f: M \rightarrow A^n$ a surjective homomorphism. Show that $Ker(f)$ is finitely generated. Following a hint in ...
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2answers
101 views

Some isomorphisms of quotient rings.

Under which conditions on the ring $A$ do we have the isomorphism $A[x]_x\cong A[x,y]/(xy-1)$, and why does this even hold? I am asking because of the following isomorphism: ...
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1answer
95 views

Ring of entire functions integrally closed or not

Is the ring $\mathscr{O}(\mathbf{C})$ of entire functions integrally closed (in its field of fractions, the meromorphic functions)? I know its not factorial, but this doesn't exclude the possibility ...
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2answers
58 views

geometry of an example regarding regularity

Consider the following situation: Let $k$ a field $X,Y$ indeterminates over $k$ and set $S=k[[X,Y]]/(Y-X^2)$ and $R=k[[y]]$, where $y$ is the class of $Y$ in $S$. Then both $S,R$ are regular but ...
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1answer
85 views

Induced map on spectrum of the rings on integral extensions

I know that if $A$ is contained in $B$ and $B$ is an integral extension of $A$, then the induced map on spectrum of the rings is surjective (and closed). Is it true if $B$ is not assumed to be ...
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1answer
40 views

A prime is minimal among primes containing an ideal

Let $I$ be an ideal in a noetherian ring $R$, and $P$ prime containing $I$. I must prove that if in the localization $R_P$, $R_P/I_P$ is annihilated by a power of $P_P$, then $P$ is minimal among ...
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1answer
61 views

Localizations of the ring $A=\Bbb{C}[X,Y]/(Y^2-X^4+2X^2-1)$

Consider the ring $A=\Bbb{C}[X,Y]/(Y^2-X^4+2X^2-1)$. Let $A_M$ denote the localization of $A$ with respect to maximal ideal $M$. My question is: Is $A_N$ a DVR, where $N$ is the maximal ideal ...
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104 views

Perfect ring Vs perfect module

In the ring theory, a left perfect ring is a type of ring in which all left modules have projective covers. The right case is defined by analogy and a ring is perfect means it is both left and right ...
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107 views

if $A^\times $ is a commutative group, does $A$ necessarily be a commutative ring?

Let $A$ be a ring and $A^\times$ be the collection of unit elements of $A$. If $A$ is a commutative ring, then $A^\times$ is a commutative group. Conversely, if $A^\times $ is a commutative group, ...
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1answer
53 views

local ring has same dimension as its regular local subring

Let $R$ be a Noetherian local ring and $S$ a regular local subring of $R$ such that $R$ is a finite $S$-module. Question: Why is it true that every regular system of parameters of $S$ is a system ...
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1answer
134 views

regarding finite integral ring extension

My question regards understanding (and possibly a source for proof) of the following, cited in the book Complex Geometry by Huybrechts (Theorem 1.1.30.) (Also, it is there stated that this is a ...
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1answer
101 views

Is the ring $\mathbb{C}[t^2,t^3]$ integrally closed?

I am trying to understand if the ring $\mathbb{C}[t^2,t^3]$ in integrally closed (into its field of fractions), but I have no idea about how to proceed. All I have tried until now has failed. Any ...
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2answers
65 views

If $N\cap rM=rN$ for all $r\in R$, then is $M=N\oplus K$ for some $K$?

Suppose $M$ is a finitely generated free module over a principal ideal domain $R$, and $N$ a submodule. Why does the condition $N\cap rM=rN$ for all $r\in R$ implies that $M=N\oplus K$ for some ...
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0answers
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Is the universal enveloping algebra of the free Poisson algebra generated by finite set (left) noetherian?

Let $P$ be the free Poisson algebra over $k$ (a field) generated by finite set $x_1,...,x_n$. Let's consider the universal enveloping algebra $P^e$ of the free Poisson algebra $P$. Hence a Poisson ...
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2answers
143 views

Dimension of quotient rings and zero divisors

In a previous question, I asked about the correctness of a method to compute the Krull dimension of quotient rings which works well if the ring in question is of the form $A/(x_1,\ldots,x_n)$, where ...
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1answer
32 views

Show that a subalgebra is commutative.

If $B$ is an unital algebra (even not commutative), how do I show that the subalgebra spanned by the elements $1$, $f$ and $(f - \lambda1)^{-1}$ is commutative? Thank you.
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1answer
86 views

Can a ring that is not finitely generated and contains $\mathbb{C}$ be Noetherian? [duplicate]

Suppose we have a ring R that contains the complex numbers, $\mathbb{C}\subset R$ and is not finitely generated as a ring. Can R be Noetherian?
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55 views

Kähler differentials of tensor product

Let $B,C$ be $A$-algebras. How can I show that $$ \Omega_{B \otimes_A C/A}=\Omega_{B/A}\otimes_A C \oplus \Omega_{C/A}\otimes_A B?$$
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1answer
72 views

Meaning of 'Isomorphism (with respect to inclusion)'

This is the first time that I see this phrase. I'm reading Commutative Algebra by N.Bourbaki. I'll extract 2 propositions that use this phrase. The first one is on page 68 of the book. ...
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1answer
59 views

Why is the tight closure tightly closed?

Let $R$ be a commutative noetherian ring containing a field of characteristic $p\gt0.$ For an ideal $I\subset R,$ the tight closure $I^*$ is defined as $$\{f\in R\mid \exists t\in R, ...
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1answer
72 views

Commutative rings with trivial automorphism group

The commutative rings $\mathbb{Z}/p\mathbb{Z}$, $\mathbb{Z}_{(p)}$, $\mathbb{Z}$, and $\mathbb{Q}$, where $p$ is a rational prime, all have trivial automorphism groups. Are there any other (unital) ...
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Krull dimension of quotient rings

This question is very related to this other question. I have an alternative solution to the ones proposed in the answers, and I'd like to know if it is correct. I want to find the dimension of ...
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1answer
59 views

If $X$ is affine reduced, show that $f\neq 0 \Rightarrow \overline {D(f)} = \operatorname {Supp} f$

If $\operatorname {Spec}A$ is reduced, show that $f\neq 0 \Rightarrow \overline {D(f)} = \operatorname {Supp} f$ Attempt at a solution: Clearly $D(f) \subset \operatorname{Supp} f$. Since the ...
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0answers
47 views

Show that $\operatorname{Spec}k[x_1,x_2,…,x_n]/(x_1^2+\cdots+x_m^2)$ is normal for $\operatorname{char}k\neq 2, n\ge m \ge 3$ [duplicate]

I want to show that if $F(T) \in B[T]$, where $B:=k[x_1,x_2,...,x_n]/(x_1^2+\cdots+x_m^2)$, is monic and has a root $\alpha \in\mathcal K(B)$ then $\alpha$ actually lives in $B$. This will imply that ...
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1answer
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Gauss lemma in UFDs

Let $A$ be a UFD, and $f\in A$ a square-free element. Define the integral domain $B:=A[z]/(z^2-f)$, and consider a monic polynomial $F(T) \in B[T]$ such that $F(\alpha) = 0$ for some $\alpha \in ...
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1answer
110 views

Poincaré series of quotient module

I am trying to calculate the Poincaré series $P(M,t)$ with respect to the standard degree grading of the graded $\mathbb{C} [x,y,z,w]$-module $ M=\mathbb{C}[x,y,z,w]/I$, where $I = (x,w) \cap (z,w) ...
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1answer
224 views

Subrings of polynomial rings over the complex plane

I have the following questions: (i) must every subring of the polynomial ring in two variables over the complex plane, containing the complex plane itself, be Noetherian? (ii) Are there Noetherian ...
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1answer
95 views

Question about tensor product of homomorphisms

I've come to think about this problem when reading a proof in Commutative Algebra by N. Bourbaki. Say, let $R$ be a commutative ring, given 3 $R-$modules $A$, $B$, $C$, and the $R$-homomorphism $f:B ...
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210 views

power series ring is faithfully flat but not free

Let $A$ be a commutative ring. Question 1: Why is the power series ring $A[[x]]$ not free over $A$ in general? Question 2: Why is $A[[x]]$ faithfully flat over $A$?
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If $x$ is integral over $A_m$ for all maximal ideals $m$, then $x$ is integral over $A$

I am going over an old exam, and there is this question that I am stuck: Given $A$ a commutative ring with unity, show that if $x\in\operatorname{Frac}(A)$ is integral over $A_m$ for all maximal ...
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66 views

can the projective dimension be read from any projective resolution?

Let $P_{\bullet}, P'_{\bullet}$ be two projective resolutions of an $R$-module $M$. Denote their differentials by $d,d'$ respectively. Define $M_i = \operatorname{ker} d_{i-1}, M'_i = ...
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1answer
118 views

homotopy equivalence of projective resolutions

Let $P_{\bullet}$ and $P'_{\bullet}$ be projective resolutions of a module $M$ over a commutative ring $R$. Then $P_{\bullet}$ and $P'_{\bullet}$ are homotopy equivalent (see e.g. Matsumura, CRT, ...
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1answer
183 views

Finish a proof that every prime ideal of a ring is the contraction of a prime ideal in its formal power series

Given a commutative ring $A$ with identity, and its formal power series ring $A[[x]]$, I am attempting to prove that every prime ideal of $A$ is the contraction of a prime ideal of $A[[x]]$. ...
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1answer
116 views

Quotient of Jacobson ring is Jacobson as in Eisenbud

I wanted to prove the following: Let $R$ be a Jacobson ring, $\mathfrak{p}<R$ a (prime) ideal. Then $R/\mathfrak{p}$ is Jacobson. The statement has been taken from Eisenbud's Commutative ...
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3answers
401 views

Counterexamples to Nakayama's Lemma if $M$ is not finitely generated

One of the most famous forms of Nakayama's lemma says: Let $I$ be an ideal in $R$ and $M$ a finitely-generated $R$ module. If $IM = M$, then there exists an $r \in R$ with $r ≡ 1 \pmod I$, ...