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

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44 views

Integral closure of $R[x]$ in its field of fractions over R

I feel like this might have been discussed before but I couldn't find it so I apologise if this is a very common question. If $S$ is a ring and we have a subring $R$ and an element $x\in S$ ...
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1answer
32 views

Difference between $\mathrm{ass}(M)$ and $\mathrm{supp}(M)$

I'm reading through Lang's chapter about Noetherian modules and rings in Algebra. More specifically, subchapter about associated primes. Let $A$ be a commutative ring, and $M$ its module. If $\...
3
votes
1answer
25 views

Length of the primary component of $(xy, y^2)$ at the origin is $1$. [on hold]

As the question title suggests, how do I see that the length of the primary component of $(xy, y^2)$ at the origin is $1$?
0
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1answer
59 views

A is an affine K-algebra and f a non-zero divisor of A. Can one say that $\dim A=\dim A_f$?

Let $A$ be an affine $K$-algebra and $f$ be a non-zero divisor of $A$ then can one say that $\dim A=\dim A_f$ ? What I proved that if $A$ is an affine domain and $f$ is a non-zero element in $A$ ...
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0answers
35 views

Question on projective modules, a course in homological algebra

Show that if $0\rightarrow N\xrightarrow{\tau} P\xrightarrow{\epsilon} A\rightarrow 0$ and $0\rightarrow M\xrightarrow{\tau'} Q\xrightarrow{\epsilon'} A\rightarrow 0$ are exact sequences with $P,Q$ ...
2
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0answers
33 views

$I=mI$, when $I$ is not finitely generated.

Let $(R,m)$ be a commutative local ring with unit. Suppose $I$ is an ideal (not finitely generated). If $I=mI$, what can we say about $I$? If $I$ were finitely generated, then Nakayama's lemma would ...
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0answers
30 views

A non domain ring with every non-zero annihilating ideal a prime ideal has a particular form.

A non-domain ring in which every non-zero annihilating ideal is a prime ideal, is of the form $F_1 \bigoplus F_2$, $F_1$, $F_2$ are fields or has only one non-zero proper ideal. Note: Here, an ideal ...
3
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0answers
56 views

Part (a) of Exercise 3.4 of Eisenbud's Commutative Algebra

In the part (a) of Exercise 3.4, it suggests that we may use the relation: $${\text{Content}(fg)}\subset{\text{Content}(f)\text{Content}(g)}\subset{\text{rad}\left(\text{Content}(fg)\right)}$$ to ...
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votes
2answers
208 views

When does tensor product have a (exact) left adjoint?

Let $A$ be a commutative Noetherian ring, and let $F$ be a flat $A$-module. We can assume $A$ is local, so $F$ is projective. Question 1. When does $F\otimes_A-$ preserve injective objects? ...
4
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0answers
67 views

Does there exist a non-field Noetherian domain whose field of fraction is the field of real numbers R?

Does there exist a non-field Noetherian domain whose field of fraction is the field of real numbers $\mathbb{R}$ ? Any help will be appreciated. Thanks
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1answer
60 views

Set of zero divisors is an ideal iff the ring is local [on hold]

Let $R$ be a commutative ring with unity. Show that $Z(R)$, the set of all zero divisors of $R$, is an ideal if and only if $R$ is a local ring. I have no idea for proving this. Thanks in advance!
0
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1answer
43 views

Isomorphic for free modules implies for projective [closed]

I have if $F$ is a free $R$-module then $\mathrm{Hom}(L,M) \otimes F$ is isomorphic to $\mathrm{Hom}(L,M \otimes F)$. Then how can I conclude that $\mathrm{Hom}(L,M) \otimes P$ is isomorphic to $\...
4
votes
2answers
70 views

Classify all schemes of degree $2$ and $3$ over $\mathbb{R}$ supported at the origin in $\mathbb{A}_\mathbb{R}^2$

Classify all schemes of degree $2$ and $3$ over $\mathbb{R}$ supported at the origin in $\mathbb{A}_\mathbb{R}^2$. In particular, show that while any such scheme $X$ whose complexification $X \times_{\...
3
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1answer
57 views

Is there a theory of generalized eigenvectors over commutative rings?

Brown's Matrices over Commutative Rings book discusses the theory of eigenvalues, eigenvectors, and diagonalizing matrices over commutative rings, but unless I've missed something, nothing like ...
1
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1answer
33 views

Krull dimension of a quotient by a system of parameters

Let $(A,\frak m)$ be a local Noetherian ring and let $x_1,\dots,x_d$ be a system of parameters, i.e. ${\frak m}=(x_1,\dots,x_d)$. Then $$\dim A/(x_1,\dots,x_i)=d-i$$ for each $i=1,\dots,d$. I know ...
2
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1answer
34 views

What does additive mean in “additive basis” in algebraic geometry?

Let $Gr(k,n)$ be the set of all $k$-dimensional subspaces of an $n$-dimensional vector space. I heard that people say "an additive basis" of $\mathbb{C}[Gr(k,n)]$. What does additive mean? Thank you ...
4
votes
1answer
43 views

$R$ be a Noetherian domain , $t\in R$ be a non-zero , non-unit element , then is it true that $\cap_{n \ge 1} t^nR=\{0\}$?

Let $R$ be a Noetherian domain, $t\in R$ be a non-zero, non-unit element, then is it true that $$\bigcap_{n \ge 1} t^nR=\{0\} \text{?} $$ It almost feels like the nilradical (which is zero for any ...
1
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2answers
113 views

Nilpotent or non-Nilpotent Jacobson Radical

Let $R$ be a ring with identity element such that every ideal of which is idempotent or nilpotent. Is it true that the Jacobson radical $J(R)$ of $R$ is nilpotent? If $R$ is Noetherian and $J(R)$ is ...
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4answers
152 views

Survey articles in Commutative/Homological algebra

I am a graduate student interested in Commutative algebra/Homological algebra. I am comfortable with first eight chapters of Atiyah. I am familiar with some algebraic geometry, first two chapters of ...
15
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1answer
780 views

Class group of $k[x,y,z,w]/(xy-zw)$

I had a homework problem (II.6.5 in Hartshorne) to compute the (Weil divisor) class group of $X=\operatorname{Spec} k[x,y,z,w]/(xy-zw)$. I have accomplished this; however, I used some results I don't ...
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3answers
37 views

$(R,\mathcal m)$ be a Noetherian local ring and let $P$ be a prime ideal of $R$. If $P^2$ is a prime ideal of $R$, then $P=0$

Let $(R,\mathcal m)$ be a Noetherian local ring and let $P$ be a prime ideal of $R$. If $P^2$ is a prime ideal of $R$, then $P=0$. I was thinking to use Nakayama lemma as: $R_P$ is local with $PR_P$ ...
1
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1answer
34 views

Idempotents in coordinate ring

Let $k$ be an infinite field. Let $V$ be an algebraic set of $k^n$. Let's suppose that $f(x)+I(V)\in k[x_1,\cdots,x_n]/I(V)$ $(x=(x_1,\cdots,x_n))$ is such that $f(x)^2+I(V)=f(x)+I(V)$. Then does $f(x)...
4
votes
1answer
61 views

Geometric differences between $\operatorname{Spec}\mathbb{C}[x]/(x^2-x)$ and $\operatorname{Spec}\mathbb{C}[x]/(x^3-x^2)$

As far as I can tell, the topological spaces associated to the schemes in the title are both sets with two elements, with the discrete topology since both have prime ideals $(x)$ and $(x-1)$ which are ...
0
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0answers
28 views

finitely generated projective R-modules

Let $L$, $M$ and $N$ be $R$-modules. Then I know that there is a natural homomorphism from $Hom(L,M) \otimes N \to Hom(L,M \otimes N)$ defined by $f \otimes n \to \tilde{f}$ where $\tilde{f}(\ell)=f(\...
0
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2answers
29 views

Principal Ideal Domain and Factorization

If $A$ is a local domain such that each non-trivial ideal factors uniquely into primes then does it follow that $A$ must be a principal ideal domain?
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3answers
26 views

Minimal ideal in a ring which is generated by an idempotent element.

Let $R$ be a commutative ring with unity and $M$ be a minimal ideal of $R$ such that $M = Re$ where $e$ is an idempotent element in $R$. Then $R = Re \oplus R(1-e) $ I am not able to see, in order ...
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2answers
46 views

What are the conditions needed for two principal ideals of a ring to be isomorphic?

Given a commutative ring $R$, and $p(x),q(x) \in R[x]$ monic polynomials, under what conditions on $p(x)$ and $q(x)$ are the principal ideals $\langle p(x) \rangle$ and $\langle q(x) \rangle$ ...
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1answer
25 views

Given a commutative ring $R$ and a monic polynomial $p(x) \in R[x]$ is $R[x]/\langle p(x) \rangle$ always a finite integral extension of $R$?

I suspect this to be true based on the fact that $p(x)$ is monic, so it should be the case that $R[x]/\langle p(x) \rangle$ is a finitely generated module over $R$, but I have no good reference for ...
1
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1answer
34 views

Minimal graded free resolution of the ideal $(x^3,xy^2,y^5)$

I am looking for a detailed explanation of every step of the construction of a graded free resolution of the ideal $(x^3,xy^2,y^5) \subseteq S=K[x,y]$ where $K$ is an arbitrary field. I saw several ...
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0answers
31 views

on the proof of Theorem 2.11 of Ratliff's “On prime divisors of $I^n$, $n$ large”

Let us consider Ratliff's paper from 1975 entitled On prime divisors of $I^n$, $n$ large, in particular the last statement of the first paragraph of the proof of Theorem 2.11. We have a Noetherian ...
2
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2answers
118 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 ...
2
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0answers
71 views

A test problem about algebraic integers in complex field

In a recent algebraic test, I meet this problem: Let R be the ring of algebraic integers in C, K is the field of algebraic numbers in C. Let a be an element of K such that the ring R[a] is ...
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0answers
46 views

Koszul complex: isomorphism between $K(a_1,\ldots, a_n;A) \simeq K(a_1;A) \otimes \cdots \otimes K(a_n;A)$

Given $a_1,\dots,a_n\in A$, with $A$ a suitable ring, my algebra teacher defined the Koszul complex associated to $a_1,\dots,a_n$ with coefficients in $A$ in this way: $$K(a_1,\dots,a_n;A):=\...
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0answers
30 views

Minimal injective resolutions isomorphism [closed]

How can I prove that given an $A$-module $M$ two injective resolutions of $M$ are isomorphic as complexes? Thank you, have a nice day Asdrubale
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1answer
49 views

Is it $\prod_{i \in \Delta } (S^{-1}A_{i}) \cong S^{-1} \prod_{i \in \Delta }A_{i}$ true? [closed]

Suppose that $A$ is a ring and $S$ is a multiplicative closed subset of $A$. I wonder that do we have $\prod_{i \in \Delta } (S^{-1}A_{i}) \cong S^{-1} \prod_{i \in \Delta }A_{i}$? Can you give me ...
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0answers
42 views

translation of “Der kanonische Modul …”

Do you know a note that is the translation of the following in English? J. HERZOG et al., "Der kanonische Modul eines Cohen-Macaulay-Rings," Lecture Notes in Mathematics No. 238, Springer-...
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2answers
64 views

Is a finite inverse limit of noetherian rings noetherian?

Let $\{A_i\}$ be an inverse system of (commutative, unital) Noetherian rings with a finite index set. Is $\varprojlim A_i$ also a Noetherian ring?
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1answer
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Example of a non-Kummer totally tamely ramified Galois extension

Let $A$ be a DVR with fraction field $K$, and let $L$ be a totally tamely ramified finite Galois extension of $K$ of degree $e$ - ie, the integral closure $B$ of $A$ in $L$ is a DVR with ramification ...
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33 views

Show that if $f_{M}: L_{M} \to G_{M}$ is surjective for every maximal ideal $ M$ of $R$ then $f$ is surjective.

Let $f:L\to G$ be a homomorphism of modules over commutative ring $R$. Show that if $f_{M}: L_{M} \to G_{M}$ is surjective for every maximal ideal $ M$ of $R$ then $f$ is surjective.
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1answer
33 views

Why is the Rees Algebra Noetherian if the underlying ring is?

Let $R$ be a commutative ring with $1$, $I \subset R$ a proper ideal. The Rees Algebra, with respect to $I$, is defined: $R[It]= \bigoplus_{n=0}^\infty I^nt^n \subseteq R[t]$. In many places I've read ...
0
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1answer
59 views

Obtain dimension of multivariate polynomial quotient ring?

Let $\mathbb{C}[z_1,z_2,...,z_n]$ be the ring of multivariate polynomials in complex variables $z_1,z_2,...,z_n$ with complex coefficients. This ring is spanned by the countably infinite basis of ...
3
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1answer
40 views

Hartshorne Prop I.4.3 Proof

$\textit{The proposition}$: On any variety $Y$, there is a base for the topology consisting of open affine sets. $\textit{The proof}$: Assume $Y$ is quasi-affine in $\mathbb{A}^n$ and let $Z=\...
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21 views

If $N$ is finitely generated, then $(L :_{R} N)^{e}= (S^{-1}L :_{S^{-1}R} S^{-1}N)$ [duplicate]

I have question in this lemma. Please help me explain it more. Let $L$, $N$ be submodules of a module $M$ over a commutative ring $R$ and let $S$ be a multiplicatively closed subset of $R$. If $N$ is ...
0
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1answer
42 views

Prime ideals in a quotient of a DVR

Suppose $R$ is a DVR. So $R$ has two prime ideals - $(0)$ and $(p)$ ($p$ the uniformizer of the maximal ideal). All other ideals in $R$ are powers of $(p)$, i.e. of the form $(p^k), k\geq 2$. I'm ...
14
votes
2answers
579 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, ...
6
votes
1answer
829 views

Tensor product of reduced $k$-algebras must be reduced?

Let $A$, $B$ be two reduced $k$-algebras. Then if an element of the form $$\sum a_{i}\otimes b_{j}$$ is nilpotent, we can compose it with any $k$-homomorphism $f$ from $A$ to $k$ to get a homomorphism ...
2
votes
0answers
60 views

Prime ideal of a polynomial ring in 6 variables

Let $k$ be a field and $k[x_1,x_2,x_3,y_1,y_2,y_3]$ a polynomial ring in 6 variables over $k$. How to prove that the ideal $(x_1y_2-x_2y_1,x_2y_3-x_3y_2,x_3y_1-x_1y_3)$ is prime in $k[x_1,x_2,x_3,y_1,...
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0answers
24 views

Separable but not reduced? [duplicate]

Say a $\Bbbk$-algebra is separable if $L\otimes _\Bbbk A$ is reduced for every field extension $L/\Bbbk$, and reduced if its underlying ring is reduced. Separable always implied reduced, and I found ...
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0answers
24 views

Rings and modules-ideals and submodules

I am taking a course in Commutative Algebra and the following lemma in a section on localisation raised some questions. Lemma: Let M be an R-module. The following are equivalent. (1) $M=0$ (2) $M_P=...
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0answers
35 views

Finite dimensional separable algebra is étale

Say a $\Bbbk$-algebra is separable if $L\otimes _\Bbbk A$ is reduced for every extension $L/\Bbbk$. Say it's étale if there's an extension $L/\Bbbk$ such that $L\otimes_\Bbbk A\cong \prod_1^nL$. Here'...