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

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3
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1answer
83 views

Characterization of free modules

Let $M$ be a finitely generated module over a commutative ring $A$. Is it true that if there exists a positive integer $n$ and a pair of homomorphisms $\pi:A^n\rightarrow M$ and $\phi:A^n\rightarrow ...
1
vote
2answers
120 views

ideal in the ring of smooth functions

What is an ideal $I$ of the ring of smooth functions $C^{\infty}(\mathbb R)$ which is not finitely generated and for all $x\in\mathbb R$ there exist $f\in I$ such as $ f(x)\neq 0$.
3
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0answers
81 views

Power series modulo polynomials

I apologize for the lengthy introduction. It is mainly for context and to introduce a certain phenomenon. $\newcommand{\Z}{\mathbb{Z}}$ Consider the groups $\Z[[x]]$ of formal power series and ...
1
vote
1answer
85 views

Semi-simple commutative algebra

Let $A$ be a semi-simple commutative algebra over a field $F$, and $F$ is algebraically closed. The proposition is that we can express $A=Fe_1 \oplus ... \oplus Fe_n$, where $e_i$ are orthogonal ...
1
vote
2answers
237 views

Ideals-algebraic set

Notice that in $\mathbb{C}[X,Y,Z]$: $$V(Y-X^2,Z-X^3) = \{ (t,t^2,t^3) \mid t \in \mathbb{C}\}$$ In addition, show that: $$I(V(Y-X^2,Z-X^3)) = \langle Y-X^2,Z-X^3 \rangle$$ Finally, prove that the ...
2
votes
1answer
80 views

Density of maximal spectrum

It's well known that for algebraically closed field $k$ maximal spectrum of finitely generated $k$-algebra is everywhere dense in whole spectrum of this algebra. What can be said in the case of ...
3
votes
0answers
100 views

A certain natural map between Tor functors

Consider the following Here $A$ is a flat (commutative, unital) $k$-algebra ($k$ a commutative ring) and $\mu:A\otimes_k A\rightarrow A$ is by $\mu(a\otimes b)=ab$, $\mathcal{M}$ denotes a maximal ...
1
vote
1answer
70 views

Basic algebraic geometry question (confused about conventions)

I am completely new to algebraic geometry so please bear with me. I have started going through James Milne's notes as a first reference and have been finding them quite good. I have now turned to ...
0
votes
1answer
44 views

Simultaneous congruences

Let $\mathbb K$ be a finite field and $\mathbb K[x, y]$ the polynomial ring in the commuting indeterminates $x$ and $y$. Consider the factor ring $\mathbb K[x, y]/\langle x^3, y^3\rangle $. Can we ...
3
votes
1answer
280 views

Other proof for tensor product equal to zero (without use Nakayama lemma and Jacobson radical)

Let $R$ is commutative local ring, $M$ is $R$-module, $N$ is $R$-module. If $M,N$ are finitely generated, how to prove: $M \otimes_R N=0$ if and only if $M=0$ or $N=0$. If we delete the ...
1
vote
1answer
54 views

Prime ideal $P$ in $R$ coprime to the conductor plus the localization $R_{P}$ is a DVR implies that $P$ is invertible

I have the following situation: Let $B \subseteq B'$ be a ring extension such that $\text{Quot}(B) = \text{Quot}(B') =: K$ and $\text{dim}(B) = \text{dim}(B') = 1$ where $B'$ is a Dedekind domain. ...
0
votes
1answer
79 views

How to check if an ideal is primary

I've the ring $A=k[x,y,z]/(z^2-xy)$ and the ideal $I=(x,y)^2$. How do I check if $I$ is prime in $A$? I know that $(x,y)$ is a prime ideal as $A/(x,y)$ is isomorphic to $k[z]$ but that does not imply ...
0
votes
2answers
117 views

Intersection of two flat submodules

Let $A$ be a ring, $M$ an $A$-module and $M_1,M_2$ two flat $A$-submodules of $M$. Is $M_1 \cap M_2$ a flat $A$-submodule of $M$?
3
votes
1answer
99 views

If $\mathfrak a\subset A$ is a finitely generated ideal, and if $\mathfrak p$ is a prime ideal, then $S(\mathfrak a)\cap\mathfrak p\not=\emptyset ?$

For an ideal $\mathfrak a\subset A,$ define $S(\mathfrak a)=\{f\in A\mid f\not\in x, \forall x\in D(\mathfrak a)\};$ namely, $S(\mathfrak a)$ is the set of elements that do not belong to any prime ...
1
vote
0answers
124 views

Counterexamples in $R$-modules products and $R$-modules direct sums and $R$-homomorphisms (Exemplification)

Q: (Exemplification) Do examples family of $R$-modules like $\{M_i\} , \{N_i\}$ and $R$-modules like $M,N$ such that every of four question in underline is true. (in other word for every question ...
2
votes
1answer
113 views

On Bounded Index of Nilpotency of $R[x]$ and $M_n(R)$

A ring $R$ is said to have a bounded index (of nilpotency) if there is a positive integer $n$ such that $x^n=0$ for every nilpotent $x∈R$. Can anyone give me an example of a ring $R$ which has a ...
1
vote
1answer
58 views

Given an ideal of a ring $R$, is there any way by which the associated primes of $R/I$ can be computed without knowing a primary decomposition of $I$?

Suppose I've been given an ideal $I$ of a commutative ring $R$ and I don't know the primary decomposition of $I$. How do I find the associated primes of $R/I$? Please give some approach if possible. ...
0
votes
1answer
40 views

Inversion of an element in Picard group over commutative ring

I'm having some troubles understanding a proof in Commutative Algebra Chapter I - VII of N. Bourbaki. It's on pag 114 of the book. Here's what it says: Theorem 3 ... (ii) Conversely, if $M$ ...
1
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1answer
62 views

Maximal among some ideals is prime

I am reading a lemma on noetherian integral domains but I am stuck, I am bring it up here hoping for help. The original passage is in one big fat paragraph but I broke it down here for your easy ...
8
votes
3answers
292 views

Are Dummit and Foote making a mistake in proving Cohen's theorem?

Exercise 11 on page 669 (this is Chapter 15) wants to prove Cohen's theorem that if every prime ideal of a ring is f.g. then every ideal is f.g. that is the ring is noetherian. The highbrow (perhaps?) ...
0
votes
0answers
37 views

$S^{-1}R[(x_i)_{i\in I}]=(S^{-1}R)[(x_i)_{i\in I}]$

Behold any commutative ring $R$. Is it true that $S^{-1}R[(x_i)_{i\in I}]=(S^{-1}R)[(x_i)_{i\in I}]$ for any multiplicative subset $R$ of $S$? I couldn't find this in full Bourbaki generality, not ...
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0answers
65 views

Calculating the Hilbert polynomial of a principal ideal

If we have a field $K$, and a homogeneous polynomial $f \in R=K[x_1, \ldots, x_n]$, then the ideal generated by $f$ is a graded module over $K$, and we can calculate its Hilbert polynomial. (I am ...
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votes
1answer
174 views

Tensor product of the fraction field of a domain and a module over the domain

Given a fraction field $k(x)$ of the polynomial ring $k[x]$ over a field $k$ and an integral domain $R$ that is also a $k[x]$-module, is it true that $k(x) \otimes_{k[x]} R \cong Frac(R)$? I ...
2
votes
1answer
73 views

Cohen-Macaulay ring and module: R-regular vs M-regular

Let $R$ be a Cohen-Macaulay ring and $M$ be a finite generated maximal Cohen-Macaulay module. I know that the R-regular sequence must be $M$-regular. Here are my questions: 1) Must an $M$-regular ...
0
votes
1answer
53 views

On Prime and Maximal Ideals in a Commutative Ring with Unity

Let $R$ be a commutative ring with $1 \neq 0$, $I$ and $P$ are ideals of $R$. If $P$ is prime and $I \cap P \neq 0$, does it follows that either $I \subseteq P$ or $I$ is also a prime ideal ...
5
votes
2answers
131 views

Can the Kahler differentials of a “good” local ring R be free of rank not equal to dim(R)?

Let $R$ be a local ring containing a field isomorphic to its residue field $k$. Assume $R$ is a localization of a finitely-generated $k$-algebra. Can $\Omega_{R/k}$ be free of rank $r\neq\dim{R}$? ...
0
votes
1answer
65 views

Dedekind domain necessary for equivalence of flatness and torsion-free

It is well-known that for finitely generated modules over a Dedekind domain, flatness and torsion-free are equivalent. Is this true for general Noetherian rings? If not, where is the dimension one ...
4
votes
2answers
97 views

Product of ideals for Nakayama's Lemma

The result to be proved is the following: Let $R$ be a local Noetherian ring. Then the minimum number of generators of the unique maximal ideal $P$ equals the dimension of $P/P^2$ as a vector space ...
2
votes
1answer
54 views

System of polynomial equations and Nullstellensatz

Let $k$ be an algebraically closed field and the field $K$ contains $k$. I am trying to prove that if $F_1,...,F_m\in k[x_1,...,x_n]$ and the system of polynomial equations $F_1=0,...,F_m=0$ has a ...
2
votes
1answer
278 views

Integral closure in field of fractions.

Let $I$ be the ideal generated by $2xy+x^2+y^3$ in $\mathbb{R}[x,y]$. Define $A:=\mathbb{R}[x,y]/I$, I want to find the normalisation of $A$, that is, the set $B= \{ a \in \text{Frac} A : \text{a ...
2
votes
2answers
555 views

Commutative ring is semisimple iff it's isomorphic to a finite direct product of fields.

I am trying to prove the following: Let $R$ be a commutative ring. Prove that $R$ is semisimple if and only if it is isomorphic to a direct product of a finite number of fields. Suppose $R$ is a ...
4
votes
0answers
72 views

a subtle detail in the proof of Theorem 3.3.7 of Bruns and Herzog

Let $\phi: (R,m,k) \rightarrow (S,n,l)$ be a local homomorphism of Artinian rings, with $k,l$ being the corresponding residue fields. Let $E_R(k)$ be the injective hull of $k$ over $R$ and $E_S(l)$ ...
2
votes
1answer
70 views

If $A$ is a semilocal ring and $f:A\rightarrow B$ is a surjective homomorphism, then $f(\operatorname{rad}A)=\operatorname{rad}B$

If $A$ is a semilocal ring and $f:A\rightarrow B$ is a surjective homomorphism, then $f(\operatorname{rad}A)=\operatorname{rad}B$. I know that if $A$ is a semilocal ring and if $I_{1},\dots, ...
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0answers
56 views

Is there any relationship between localization and completion of a module?

Let $R$ be a commutative ring, $\mathfrak p$ a prime ideal of $R$ and $M$ an $R$-module. I've seen the terms 'localization' $M_\mathfrak p$ of $M$ and the completion $M_\mathfrak p$ at $\mathfrak p$ ...
3
votes
1answer
42 views

valuation ring, completeness

Perhaps a trivial question: is there an example of a field $K$ and a valuation $v$ on $K$ such that the following holds: $K$ is not complete (with respect to the valuation topology) The valuation ...
4
votes
3answers
241 views

Recommendations for Commutative Algebra Software?

I'd like a software that I can use to work with commutative algebra, specifically to figure out S-Polynomials, Buchberger's Algorithm, etc. I have Mathematica; if anyone could refer me to a package, ...
2
votes
0answers
111 views

Noetherian normal ring is a finite direct product of normal domains

Let $A$ be a Noetherian normal ring, that is, the localization of $A$ at every prime is a normal domain. I want to show $A$ is a finite product of normal domains. If $p_1,\ldots,p_n$ are the ...
0
votes
1answer
159 views

Why does a ring homomorphism induce a continuous map between spectra? [duplicate]

Let $\varphi: A \rightarrow B$ be a ring homomorphism. Let $f =\mathrm{Spec}(\varphi) : \mathrm{Spec}(B) \to \mathrm{Spec}(A)$ be the map associated to $\varphi$. Why is the map $f$ is continuous? ...
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0answers
30 views

Improvement of Buchberger's Algorithm (second part)

Suppose $S_j$ is a homogeneous syzygy of multidegree $\gamma_j$ in $S(G)$, where $G=\{g_1,\dots,g_t\}$. Show that $S_j G=\Sigma_{i=1}^{t} c_ix^{\alpha(i)}g_i$ has multidegree $< \gamma_j$. Now, I ...
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vote
1answer
83 views

Question concerning the chinese remainder theorem for commutative rings

let $S$ be a commutative ring and $I_1,...,I_n\unlhd S$, such that $I_i+I_j=S\ \forall i\neq j$. Let $g_1,...,g_n\in S$. Why are there $h_1,...,h_n,h'\in S$, such that ...
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0answers
44 views

Radical of an ideal in a finitely generated ring over $k$ is the intersection of maximal ideals containing it. [duplicate]

From Matsumura p.34 Let $k$ be a field, $A$ a ring which is finitely generated over $k$, and $I$ a proper ideal of $A$; then the radical of $I$ is the intersection of all maximal ideals containing ...
2
votes
1answer
31 views

2 questions concerning identities of closed subspaces of $spec(S)$ for a commutative ring $S$

I have the following questions: Let $S$ be a commutative ring and let $M,N$ be closed subspaces of $spec(S)$, such that $M\cap N=\emptyset$. 1) Why are there ideals $I_1,I_2\unlhd S$, such that ...
2
votes
1answer
57 views

integral ring homomorphism

Consider a homomorphism $f: A\to B$ of commutative rings and let $b\in B$. Let $g\colon A[X]\to B[X]$ be defined by $g(X) = X$. Put $I = g^{-1}((bX-1))$ (contraction of the ideal $(bX-1)\subseteq ...
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0answers
20 views

Inverse image of a maximal ideal under a morphism of finitely generated $\mathbb{C}$-algebras. [duplicate]

Let $$ f: A\to B $$ be a morphism of finitely generated $\mathbb{C}$-algebras, suppose $\mathfrak{m}\unlhd B$ is a maximal ideal, I want to show that $f^{-1}(\mathfrak{m})$ is a maximal ideal of $A$. ...
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votes
1answer
78 views

Finding a coordinate ring

I am having hard time in calculating (or constructing) $\displaystyle\frac{\mathbb C[x,y]}{\langle y^2 - x^3 - x\rangle}$. I tried homogenizing the ideal $y^2 - x^3 -x $ to $ wy^2 - x^3 - xw^2$. But ...
0
votes
1answer
53 views

Functorial isomorphism involving tensor products

Let $R$ be a commutative ring and $E', E, F', F$ be free, f.g. $R$-modules of equal rank. For $f\in L(E',E):={\rm Hom}_R(E',E)$ and $g\in L(F',F)$, let $T(f.g)\in L(E'\otimes_R F', E\otimes_R F)$ be ...
0
votes
0answers
106 views

Irreducibility of a polynomial and connectedness of its zero set

Let $P$ be a polynomial in $\mathbb{C}[z_1,z_2,...,z_n].$ Let $Z(P)$ denotes its zero set in $\mathbb{C}^n.$ I have the following question: Does the irreducibility of $P$ imply that $Z(P)$ is ...
0
votes
0answers
81 views

Induced Spec map for a morphism of finitely generated $\mathbb{C}$-algebras

I have a morphism $f:A\longrightarrow B$ of finitely generated $\mathbb{C}$-algebras. I have proven, using Zariski's lemma, that the inverse image of a maximal ideal $M \subset B$ is a maximal ideal ...
12
votes
0answers
153 views

How to name these “ideals”?

Background. Let $\mathcal{C}$ be a symmetric monoidal category with unit $\mathbf{1}$. A subobject of $\mathbf{1}$ is just a monomorphism $I \to \mathbf{1}$. We may also call this an ideal of ...
1
vote
1answer
90 views

Saturation of a power of an ideal

Let $k$ be a field and let $R=k[x,y,z]$ and $\mathfrak m=(x,y,z)$. Let $I$ be a graded ideal of $R$. For all $n\in \mathbb{N}$ on has $$ (I^{\rm sat} )^n\subset (I^n)^{\rm sat},$$ where $$I^{\rm ...