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

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1
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0answers
24 views

If $\pi : Proj(T_*) \to Proj(S_*)$ is induced by some $S_* \to T_*$, does some tensor product describe $\pi^*(\widetilde{M_*})$

If $\pi : Proj(T_*) \to Proj(S_*)$ is induced by some $S_* \to T_*$, does some tensor product describe $\pi^*(\widetilde{M_*})$? $\pi^*(\widetilde{M_*}) =_? T_* [\otimes]_{S_*} M_*$?, where ...
4
votes
2answers
39 views

$A$ noetherian, $A$-endomorphism not injective for all invariant submodules is nilpotent

Let $A$ be a noetherian ring, $M$ a finitely generated $A$-module, $T: M \to M$ an endomorphism. Assume that for all $T$-invariant proper submodules $N$ of $M$, the induced endomorphism $\overline ...
1
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0answers
30 views

If $R$ and $S$ are two graded rings, is there a name for the construction $\oplus_{n \geq 0} R_n \otimes_{\mathbb{Z}} S_n$?

If $R$ and $S$ are two graded rings, is there a name for the construction $\oplus_{n \geq 0} R_n \otimes_{\mathbb{Z}} S_n$? This gives a graded ring, but it is not quite the tensor product since we ...
3
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2answers
42 views

If $M_*$ and $N_*$ are graded modules over the *graded* ring $R_*$, what is the definition of $M_* \otimes_{R_*} N_*$?

Quick question (hopefully): What is the correct definition of a tensor product of two graded $R_*$-modules and/or graded $R_*$-algebras $M_*$ and $N_*$ over the graded ring $R_*$? $M_* \otimes_{R_*} ...
2
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1answer
59 views

$T$ automorphism of a finitely generated $A$-module, then $T^{-1} \in A[T]$

Let $M$ be a finitely generated $A$-module. Let $T: M \to M$ be an isomorphism. Then $T^{-1}$ is a polynomial in $T$ with coefficients in $A$. This seems to me as a direct application of ...
0
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2answers
32 views

Can we always choose the generators of an ideal of a Noetherian ring to be homogeneous?

Let $R$ be a $k$-subalgebra of $S=k[x_1,x_2,\dots,x_n]$. Let $m\in R$ be the ideal generated by the homogeneous elements of $R$. As $S$ is Noetherian, the ideal $mS$ has a finite set of generators, ...
1
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1answer
26 views

Finding an irreducible polynomial

Suppose k is infinite. Then the irreducible algebraic subsets of $A^2(k)$ are: $A^2(k)$,$\emptyset$, points, and irreducible plane curves $V(F)$, where $F$ is an irreducible polynomial and $V(F)$ is ...
5
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1answer
155 views

Showing a polynomial irreducible

How to show that the polynomial $Y^2+X^2(X-1)^2$ is irreducible in $\mathbb R[X,Y]$. I tried to show that $\mathbb R[X,Y]$ modulo this ideal is an integral domain but I cannot find any homomorphism.
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1answer
43 views

Prove that $\operatorname {ht}(p/a)\leq \operatorname {ht}(p)\leq \operatorname {ht}(p/a)+n$

In the Milne's book A Primer of Commutative Algebra, pg. 100, there's a proof that $\operatorname {ht}(p/a)\leq \operatorname {ht}(p)\leq \operatorname {ht}(p/a)+n$. I understand the first inequality, ...
1
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1answer
31 views

Functor right adjoint to $.\otimes_BA$

Given a ring morphism from $B$ to $A$, we can regard an $A$-module $M$ as a $B$-module. Then how can I prove the functor $._B:\operatorname{Mod}_A \to \operatorname{Mod}_B$ is right adjoint to ...
0
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0answers
21 views

When do we have $\operatorname{Hom}_{R}(L,M\otimes N)\cong \operatorname{Hom}_R(L,M)\otimes N$? [duplicate]

Let $R$ be a commutative ring and $L,M,N$ be $R$ modules. I would like to know that when is the natural map $$ \operatorname{Hom}_R(L,M)\otimes_R N\to \operatorname{Hom}_R(L,M\otimes_R N) $$ is an ...
0
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1answer
60 views

A relation between max-spectrum and spectrum of a ring

For a commutative ring $R $ with identity we know that if the prime spectrum, the set of all prime ideal with Zariski topology, is noetherian then max spectrum, the set of all maximal ideal, is also ...
0
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1answer
44 views

Isn't a subalgebra of a finitely generated k-algebra always finitely generated? [duplicate]

Let $K[x_1, x_2,\dots, x_n] $ be a polynomial ring. If it is a graded ring, then under certain conditions, its subalgebras may be finitely generated. Isn't a subalgebra of a finitely generated ...
0
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0answers
36 views

Exercise 2.2 in Atiyah and Macdonald's Introduction to Commutative Algebra.

I am asked to prove that $(N:P) = Ann((N+P)/N)$, where $N,P$ are submodules of a module $M$, and $x\in (N:P) \Leftrightarrow xP \subset N$. I was thinking along the following lines: $$ x\in ...
0
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1answer
75 views

An inequality about the dimension of fiber

I am working on Problem 11.4.A of Vakil's notes: Let $X$ and $Y$ be two locally noetherian schemes, and $\pi:X \to Y$ is a morphism. $\pi(p)=q$. Then prove: $codim_Xp \leq ...
0
votes
1answer
25 views

Fractional ideals in the quotient field of Dedekind

Let $R$ be a Dedekind ring, $K$ its quotient field. If $J$ is a fractional $R$-ideal in $K$ then I want to show that $KJ=K$, so that it's a full $R$-lattice in $K$. Since $J$ is non-zero, we can ...
-5
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0answers
36 views

Show that this ring is graded, integral and noetherian [closed]

$R = \Bbb C[x]$ is a filtered ring with $F^i R = \{ \text{polynomials of degree } \le i\}$. Show that $Gr R = \bigoplus \limits _{i \ge 0} {F^i R}/{F^{i−1} R}$ is naturally a (graded) ring. Show ...
3
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1answer
35 views

Zariski topology on $\mathbb{C}[X, Y]$

For a commutative ring $A$, let Spec$(A)$ be the set of prime ideals. A topology on Spec$(A)$ is defined by the closed sets $$ \mathcal{V}(T) = \lbrace \mathbb{p} \in \text{Spec}(A) \vert T \subseteq ...
1
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1answer
92 views

Vakil FOAG 11.3.B

I am thinking about how to use Krull's PIT to prove this statement (11.3.B on Vakil's notes): If $(A,m,k)$ is a Noetherian local ring with maximal ideal $m$, and $f \in m$, then $\dim A/(f) \geq ...
2
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2answers
55 views

Correspondence between nilpotents and between idempotents

It is well-known and easily proved that whenever $R$ is a commutative ring with unity and $S$ is a multiplicative subset of $R$, each ideal of the localization ring $R_S$ is an extended ideal (with ...
2
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1answer
69 views

Prove that $m_1m_2\ldots m_r=n_1n_2\ldots n_s$ implies $r=s$ for distinct maximal ideals

Let $R$ be a commutative ring where $m_1,m_2,\ldots,m_r$ and $n_1,n_2,\ldots,n_s$ are maximal ideals such that $m_1m_2\ldots m_r=n_1n_2\ldots n_s$ and $m_i \neq m_j$, $n_i \neq n_j$ if $i \neq j$. ...
0
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1answer
35 views

Proving $R/J$ is local, where $R=k[\Gamma]$ and $J=(x^1)\unlhd R$.

Let $\Gamma$ be the set of symbols of the type $x^q$, where $q\in\Bbb Q, \;q\ge0$. Setting $x^{q_1}\cdot x^{q_2}:=x^{q_1+q_2}$, $(\Gamma,\cdot)$ becomes a semigroup. Let then $k$ be a field. Let's ...
2
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0answers
33 views

Can we use Atiyah-Macdonald Proposition 1.11(ii) to prove the Lagrange interpolation theorem?

Two Theorems: Theorem 1 (Atiyah-Macdonald, Proposition 1.11(ii)): Let $A$ be a commutative ring, and let $\mathfrak p_1,\dots,\mathfrak p_n$ be prime ideals in $A$, and let $\mathfrak a$ be an ideal ...
0
votes
2answers
74 views

Closed subschemes of affine scheme

In Mumford's Red Book (p. 106, 2nd edition, Theorem 3) it is proved that any closed subscheme $(f,f^\sharp):Y→X, f$ the inclusion, of an affine scheme $X=\operatorname{Spec}R$ is of the form ...
1
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1answer
27 views

Number of Associated Prime Ideals vs. Number of Maximal Ideals

This is a very naive (and presumably basic) question, but suppose you have an ideal $I \subset R$ in a commutative Noetherian ring with unity. Does the number of maximal ideals containing $I$ have ...
2
votes
1answer
68 views

For a ring $R$, what is $\text{Gr}(R)$?

I'm reading Deligne's "The fundamental group of the projective line minus 3 points", specifically the chapter on tangential base points (15.14), where in 15.20, he suddenly uses the notation $Spec\; ...
0
votes
1answer
25 views

Normalization of ring of polynomials

Let $x_1(t),...,x_n(t)\in\mathbb{C}[t]$ be such that $\mathbb{C}[t]$ is finite as a $\mathbb{C}[x_1(t),...,x_n(t)]$-module and that $\mathbb{C}(x_1(t),...,x_n(t))=\mathbb{C}(t)$. How to show that ...
1
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1answer
36 views

What is the integral closure of the integers in the real numbers?

What is the integral closure of the ring $\mathbb{Z}$ inside the field $\mathbb{R}$ of real numbers and what are it's properties? Is this studied at all?
0
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1answer
35 views

What is considered to be the natural (injective) homomorphism $\frac{IL+J}{IL} \rightarrow I \otimes \frac{R}{L}$?

Let $R$ be a ring and $I,J,L \unlhd R$ such that $J \subseteq I$. What is considered to be the natural homomorphism $\frac{IL+J}{IL} \rightarrow I \otimes \frac{R}{L}$ ? Remark: It must be ...
0
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0answers
50 views

Is it true that taking injective hull commutes with the tensor product?

Let $M$ and $N$ be two modules (can assume them to be finitely generated if need be) over the ring $A=k[x_0,...,x_n]$. Denote by $E(M)$ the injective hull of $M$. We work in the category of positively ...
2
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1answer
26 views

Commutativity of $\operatorname{End}_{R}(M)$ when $M$ is a semi-simple module

Let $M$ be a semi-simple module over a unital ring $R$. I want to see if $\operatorname{End}_{R}(M)$ is commutative only if $M$ is a direct sum of pair-wise non-isomorphic simple modules.
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2answers
31 views

If $\phi:A\to B$ is a ring homomorphism, why does there exist $\psi:\text{spec}(A)\to \text{spec}(B)$?

Let $\phi:A\to B$ be a ring homomorphism, where $A$ and $B$ are commutative rings. We know that if $q$ is a prime ideal in $B$, then $\phi^{-1}(q)$ is a prime ideal in $A$. Hence, there exists a ...
2
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0answers
59 views

Invariant ring for $S_5$ [closed]

For the standard representation of $S_5$, the ring of invariants is generated by the elementary symmetric polynomials and hence it is a polynomial ring. Now if we take the tensor product of standard ...
1
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1answer
32 views

Radical of an ideal in $R [x]$

Let $\frak {I}$ be an ideal of $R[x]$, the polynomial ring over a commutative ring with identity $R$. Is it true that the radical of $\frak{I}$, the intersection of all prime ideals containing ...
-2
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1answer
38 views

Dimension localization [closed]

Let $A$ be the localization of $\mathbb Z[x, y]$ in the ideal $(5, x−1, y+2)$ and $B = A/(x^2+y^2+4y−3x+6)$. Calculate the dimensions of $A$ and $B$ and study if they are regular rings.
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33 views

Any ideal is an extended one

It is true for any commutative rings $S$ and $T$ with $1$ and any ring homomorphism $f:S\to T$ that the set $E$ of extended ideals in $T$ equals $\{J\mid J^{ce}=J\}$. In fact, if an ideal $J$ of $T$ ...
2
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2answers
67 views

Sum of ideal sheaves commutes with taking global sections

Let $X$ be a smooth degree $d$ hypersurface in $\mathbb{P}^3$ and $C, D$ effective divisors intersecting each other at finitely many points. Is it true that ...
3
votes
1answer
24 views

Extension of idempotent ideals

Let $R$ be a Noetherian commutative ring with $1$. If $R[[x]]$ denotes the ring of formal power series over $R$ and $I$ is an idempotent ideal of $R$ I want to know whether the extension of $I$ in ...
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0answers
42 views

Computing Krull dimension of $\mathbb{Z}[X_1,\ldots,X_n]/I$ [closed]

Let $I$ be an ideal of $\mathbb{Z}[X_1,\ldots,X_n]$. How does one compute the Krull dimension of $\mathbb{Z}[X_1,\ldots,X_n]/I$? Are there any general methods? Or methods which work in special cases?
4
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2answers
88 views

Let $(R,M)$ be a local ring. Suppose that $R$ is noetherian and let $I,J \unlhd R$ such that $J \subseteq I$. Prove that the following are equivalent.

Let $R$ be a local ring with maximal ideal $M$. Suppose that $R$ is noetherian and let $I,J$ be ideals of $R$ such that $J \subseteq I$. Consider the following statements: 1) Every minimal set of ...
7
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1answer
99 views

Is $\mathbb{R}[x,y,z]/(x^2+y^2+z^2)$ a UFD?

As the title says, I am curious as to whether $A =\mathbb{R}[x,y,z]/(x^2+y^2+z^2)$ is a UFD. I believe the answer is yes. A thought I had was to apply Nagata's criterion, say by localizing ...
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0answers
66 views

Commutative algebra text that solely contains 200+ exercises

I am looking for a textbook that I came across awhile ago that I have been unable to find for the last week or so of periodic searching. The textbook had nothing other than, from memory 247 exercises ...
0
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1answer
60 views

Given $I=\langle xy, xz+z(y^2-z^2)\rangle$, prove that $I=\langle x, z(y^2-z^2)\rangle \cap \langle y, xz-z^3)\rangle $.

This is Exercise 3c. from Chapter 9, Section 7 of Ideals, Varieties, and Algorithms by Cox et al. Given $I=\langle xy, xz+z(y^2-z^2)\rangle$, prove that $I=\langle x, z(y^2-z^2)\rangle \cap ...
4
votes
1answer
56 views

Does $S = R \cap K$ of a field extension $K \subseteq L = Q(R)$ satisfy $Q(S) = K$?

If $K$ is finite field, then one can easily show that there is no proper subring $R$ with $Q(R) = K$, where $Q(R)$ is the field of fractions of $R$. As a consequence, algebraic extensions $K$ of ...
11
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0answers
118 views

If $A$ is the ring of continuous functions on a genus $g$ surface, can the genus of $X$ be seen by simple algebra in $A$? [migrated]

I was describing to a friend the result that a compact Hausdorff space is determined up to homeomorphism up to by its ring of continuous functions, and he asked how one could see the genus of a ...
0
votes
0answers
21 views

Locally presentable sheaves and the associated module functor

Let $R$ be a commutative ring. Any $R$-module has a presentation $R^{(J)}\rightarrow R^{(I)}\rightarrow M\rightarrow 0$. The associated module functor $M\mapsto \tilde M$ is exact and so preserves ...
0
votes
0answers
15 views

Counter example that Artinian k-algebra is not finite k-vector space [duplicate]

Let k be a field, A be a k-algebra. If A is not a finitely generated k-algebra,then the following two conditions are NOT equivalent: (i) A is Artinian; (ii) A is a finitely k-algebra, i.e. A is ...
3
votes
2answers
51 views

If $f: A\to B$ is faithfully flat and $B$ is an Artinian ring then $A$ is also Artinian.

Let $f : A → B$ be a map of rings. The map $f$ is called faithfully flat if $B$ is flat $A$-module ($B$ is $A$-module w.r.t. multiplication defined by $ab := f(a)b$) and if for any $A$-module $M, M ...
0
votes
0answers
52 views

Krull dimension of quotient ring

What is the Krull dimension of $B = A[x,y,z]/\langle x^2y + x + 1, y^3 + 2z + 1 \rangle $ given $A$ is a Noetherian, commutative ring? (Assuming that all coefficients are non zero in $A$)
4
votes
1answer
47 views

Maximal ideal in $\mathfrak{R}[x]$

Let $\mathfrak{R}$ be a commutative ring with identity. Show that if there exists a monic polynomial $p(x)\in \mathfrak{R}[x]$ of degree at least one such that the ideal $(p(x)) ...