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

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name of a certain class of rings

Does there exist a name for the class of commutative rings with identity that satisfy the following: For any 2 ideals $I_1,I_2$ of R,we have : $I_1 I_2= (I_1\cap I_2)(I_1+I_2) $ I would also like to ...
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0answers
50 views

What is the relation between the two following dimensions in polynomial ring?

Let ‎$‎I‎$ ‎be an ‎ideal ‎of $‎R=\Bbb K[x_1,‎\ldots‎,x_n]‎$‎‎ and ‎$‎{\bf u}=\{x_j\}_{j=1}^{\ell<n}‎\subset‎\{x_1,‎\ldots‎,x_n\}‎$‎‎‎. $‎{\bf u}‎$ ‎is called ‎independent ‎modulo ‎‎$‎I‎$ ‎if ‎‎$I\...
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1answer
69 views

When are powers of prime ideals primary?

This is a follow up to: Normal domains and powers of height one primes In the comments to the linked question, user26857 noted that the prime ideal $P = (x,z)$ in the Noetherian normal domain $k[x,y,...
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1answer
33 views

If $f:R\to S$ is a homomorphism of commutative rings and $Q$ a primary ideal of $S$, then $Q^{c}=f^{-1}(Q)$ is a primary ideal of $R$

Let $f:R\to S$ be a homomorphism of commutative rings, and let $Q$ be a primary ideal of $S$.Then $Q^{c}=f^{-1}(Q)$ is a primary ideal of $R$. I wonder if $f(R)=Q$ then what happens? I suppose that $...
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1answer
44 views

Intersection of height one localisations of a normal Noetherian domain

Exercise 8.3 in Kemper's A Course in Commutative Algebra: Let $R$ be a commutative normal Noetherian domain. Prove that $$R=\bigcap_{\substack{P\in\operatorname{Spec}(R)\\\operatorname{ht}(P)=1}}...
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22 views

A concrete example of an ideal $I\subseteq K[x_1,\ldots,x_n]$ and coprime polynomials $f,g$ such that $(I,f)\cap (I,g)\neq (I,fg)$

I know that in a polyomial ring $K[x_1,\ldots,x_n]$ over a field $K$, given a monomial ideal $I$ and two coprime monomials $f,g\notin I$, it holds $$(I,f)\cap (I,g)=(I,fg)$$ However, I've been ...
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1answer
46 views

If $X(F) \cap Y$ is dense in $Y$, then $Y$ is defined over $F$.

Let $k$ be an algebraically closed field, $F$ a subfield of $k$, $A$ a finitely generated, reduced $k$-algebra, and $A_0$ an $F$-subalgebra of $A$, of finite type over $F$, such that the canonical $k$-...
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1answer
76 views

Homological dimension of categories of modules

Let $A$ be a Noetherian ring. We have two categories: (a) category of $A$-modules (b) category of finite type $A$-modules. Do their homological dimensions agree? The homological dimension of an ...
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1answer
43 views

When $(f(T),f'(T))=R[T]$?

Let $R$ be a UFD, $f(T) \in R[T]$ a monic polynomial of degree $d \geq 2$, and $f'(T)$ the formal derivative of $f(T)$. When the ideal generated by $f$ and $f'$ equals $R[T]$? (If $d=1$, then $f'=1$,...
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2answers
58 views

Smooth algebras

All rings are Noetherian and eft. An $A$-algebra $B$ is smooth if it is flat and the fibres are geometrically regular. I want to see some examples of this notion. So I considered the $\mathbb Z$-...
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2answers
106 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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0answers
62 views

Normal domains and powers of height one primes

Let $A$ be a Noetherian normal domain, and $P$ a height 1 prime of $A$. Then the localization $A_P$ is a discrete valuation ring. Certainly $PA_P\cap A = P$. Does this also hold for higher powers of $...
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1answer
202 views

I don't get ring theory. What am I doing wrong? Please help. [closed]

Please allow me to ramble a bit. All my rings are commutative with $1$. I've done two semester's worth of commutative algebra; in particular, a 3rd year undergraduate subject called "Rings, Modules ...
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1answer
40 views

Generator of intersection of ideals in a PID via adjunction?

In a PID we have the formulas $ \left\langle f\right\rangle + \left\langle g \right\rangle = \left\langle \gcd(f,g) \right\rangle $ and $ \left\langle f\right\rangle \cap \left\langle g \right\rangle =...
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41 views

Special cases of prime avoidance theorem

Let $\{p_i\}$ be a family of minimal prime ideals in a commutative ring $R$ with $1$, and let $I$ be a finitely generated ideal of $R$ such that $I\subseteq \cup p_i$. Can we deduce that there exist $...
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31 views

Every irreducible submodule is primary

Sea $R$ un anillo conmutativo con identidad noetheriano y $M$ un $R$-módulo finitamente generado. $N$ es un submódulo propio de $M$. Entonces si $N$ es irreducible implica que $N$ es primario. ...
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2answers
90 views

When is $X_1^{a_1} \cdots X_n^{a_n}-1$ irreducible?

Let $F$ be a field, and $a_1, ... , a_n \geq 1$ integers. When is the polynomial $$f = X_1^{a_1} \cdots X_n^{a_n}-1$$ irreducible in $F[X_1, ... ,X_n]$? I believe this should be the case if and ...
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1answer
45 views

$(x_1, …, x_k)$ is prime in $R[x_1, …, x_n]$ if $R$ is an integral domain

Let $R$ be an integral domain. I need to prove that $\forall k = 1, ..., n \ \ \ (x_1, ..., x_k)$ is prime in $R[x_1, ..., x_n]$. I managed to do it for $k = 1$. Let $f, g \in R[x_1, ..., x_n]$. Then ...
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1answer
48 views

Does localization commute with taking radicals?

Let $A$ be a ring, $S\subset A$ a multiplicative set, and $I\subset A$ an ideal not intersecting $S$. For any ideal $J$, let $r(J)$ denote the radical of $J$. Is $S^{-1}r(I) = r(S^{-1}I)$? ...
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39 views

Proof of Krull's intersection theorem with Taylor expansion

I took a commutative algebra course last semester (with Kaplansky's book), and I've learned about Krull's intersection theorem. In the course, we proved it without using Artin-Rees Lemma. I heard that ...
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12 views

Basic fibres for monomials

The following definitions are from Irena Peeva's book Graded Syzygies. Let S = k[x$_1$,...,x$_n$], the set of all monomials in S of multidegree $\alpha$ is called the fibre of $\alpha$. We denote gcd(...
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1answer
59 views

If $p$ is a prime ideal then $p[X]$ is a prime ideal

If $Z$ is a ring and $p$ is a prime ideal of $Z$ then $p[X]$ is a prime ideal of $Z[X]$. Is it true or false? I believe that it is true and I try to prove it like that: Take $f(x)\in p[X]$ and ...
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24 views

Let $(f_1,\ldots,f_r)\subset k[x_1,\ldots,x_n],(g_1,\ldots,g_s)\subset k[y_1,\ldots,y_m]$ be radical ideals. Then their sum is radical. [duplicate]

Let $\mathfrak{a}:=(f_1,\ldots,f_r)\subset k[x_1,\ldots,x_n],\mathfrak{b}:=(g_1,\ldots,g_s)\subset k[y_1,\ldots,y_m]$ be radical ideals. Then I wish to prove that $\mathfrak{c}:=(f_1,\ldots,f_r,g_1,\...
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3answers
81 views

Form of the elements of a localization

If I have a ring $R$, a multiplicatively closed subset $U\subset R$, and consider an element of the localization: $\frac{r}{r'} \in U^{-1}R$, can I then assume without loss of generality that $r'\in U$...
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1answer
56 views

Is every unramified extension of DVRs simple?

Let $A$ be a discrete valuation ring with maximal ideal $\mathfrak{m}$, fraction field $K$, and $L$ a finite separable extension of $K$ degree $n$, unramified w.r.t. $A$. Let $B$ be the integral ...
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1answer
32 views

The set of prime ideals, whose contraction is a fixed prime in the ring of invariants, is finite.

Let $A$ be a domain and $G$ a finite group of automorphisms of $A$. I define $$A^G=\{a\in A\mid\sigma(a)=a ,\forall\sigma\in G\}.$$ Furthermore let $S\subset A$ be multiplicatively closed such that $\...
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1answer
19 views

Group action on algebra over a field defined on generators

Suppose $G$ is a group and $A$ is a finitely generated algebra over a field $\mathbb{k}$. Let $X=\{x_1,...,x_n\}$ be a set of generators for $A$, and suppose $G$ acts on $X$. Is this enough to define ...
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2answers
65 views

Shouldn't $t^n : \mathbb{A}^1 \rightarrow \mathbb{A}^1$ ramifies at $0$?

Yo, this is probably the stupidest question ever that I've asked here. Let $$\varphi: \mathbb{A}^1 \rightarrow \mathbb{A}^1$$ be the map of schemes (over a field $k$) such that $\varphi (x) = x^n$. ...
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1answer
46 views

Localizing a ring of invariants

Let $A$ be a domain and $G$ a finite group of automorphisms of $A$. I define $$A^G=\{a\in A\mid\sigma(a)=a ,\forall\sigma\in G\}.$$ Furthermore let $S\subset A$ be multiplicatively closed such that $\...
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0answers
39 views

How can I compute completions of rings?

I want to learn about how to compute the completions of local rings. For example, I want to be able to compute the completions of \begin{align*} \left(\frac{\mathbb{C}[x,y]}{(y^2 - x)}\right)_{(x,y)} ...
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1answer
45 views

Isomorphism of modules [duplicate]

Are $\mathbb{C}[x,y]/(x,y)$ and $\mathbb{C}[x,y]/(x-1,y-1)$ isomorphic as $\mathbb{C}[x,y]$-modules? I think they are cyclic so they are isomorphic, but I'm not sure.
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1answer
43 views

Faithfully flat descent of projectivity and freeness

I am reading this paper. It is proven there that if $f:A\rightarrow B$ is a faithfully flat morphism of rings and $M$ an $A$-module such that the $B$-module $M\otimes_A B$ is projective, then $M$ ...
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0answers
30 views

Homogeneous System of Parameters

Assume that $R$ is a finitely generated graded $k$-algebra of Krull dimension $n$. Is it true that any set $\{f_{1},f_{2},...,f_{n}\}$ of homogeneous algebraically independent polynomials is a ...
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1answer
22 views

A condition that the ratio of locations is maximal

Sea $R$ un anillo conmutativo con identidad e $I$ un ideal de $R$ y $m$ un ideal maximal de $R$. Mostrar que $\displaystyle\frac{R_m}{I_m}\neq{0}$ si y solo si $I\subseteq{m}$. Dm: $[\Rightarrow{}]$. ...
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24 views

Combinatorial commutative algebra

Let G is simple graph and Δ(G) is clique complex, IΔ(G) has a 2-linear resolution if and only if, for any subset W ⊂ [n], one has H˜i(Δ(G)W ; K) = 0 unless i=0 (from comment): I do not understand ...
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38 views

Find an $R$-module homomorphism $f:R\longrightarrow M$ such that $r_0m=0$ implies $r_0f^{-1}(m)=0$ , $(m\in M)$

Let $R$ be a commutative ring with $r_0\in R$ a fixed element, and $M$ be an $R$-module. I search for an $R$-module homomorphism $f:R\longrightarrow M$ such that $r_0m=0$ implies $r_0f^{-1}(m)=0$ , $(...
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2answers
34 views

Maximal ideal in a local artinian ring.

I know that an artinian ring $A$ is the union of its units and its zero-divisors. So every non-zero-divisor is an unit. I also know that in a local ring every element which is out from the maximal ...
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2answers
64 views

Finite commutative ring with unity and without nilpotent elements

Let $R$ be a commutative ring with unity such that for each $x \in R$ there exists a $n \in \mathbb{N}$, $n>1$, such that $x^n = x$. Then show that $$ R\simeq F_{1}\times F_{2}\times \cdots\times ...
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1answer
31 views

about minimal prime ideals [closed]

Let $R$ be a ring with minimal prime ideals $p_1,\ldots, p_n$ and $D=R/{p_1}\times \cdots \times R/p_n$. Please find an element $x\in R$ such that $\mathrm{ann}_D(x+p_1,\ldots,x+p_n)=\mathrm{ann}_D(1+...
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51 views

Hartshorne's algebraic geometry ; geometric understanding and intuition for intersection multiplicity

I am reading section $7$ of the book. He defines intersection multiplicity as Let $Y$ be a projective variety of dimension $r$. Let $H$ be a hypersurface not containing $Y$. Then by (7.2) $Y\cap ...
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38 views

Understanding the Definition of minimal prime ideal of a graded module

I am reading algebraic geometry from Robin Hartshorne. He has used a term "$p$ is a minimal prime of a graded $S$ module $M$". What does it mean? I know the definition of minimal prime over an ideal.
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1answer
41 views

Noetherian vector space is finite-dimensional

Given a field $k$, and a $k$-vector space $V$ which is noetherian as $k$-module, I want to show that $V$ is finite-dimensional. Is it correct that this follows because since $V$ is noetherian, every ...
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1answer
39 views

Let $M_1$, $M_2$ be Artinian modules over $R$. Then $M_1\times M_2$ is Artinian.

Using exact sequences, it's fairly easy to prove the converse, but I can't figure out how to prove this statement. Suppose we have a descending chain $N_1\supset N_2\supset\cdots$ of $R$-submodules ...
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1answer
54 views

The ideal of the image of homogeneous polynomials

Let $k$ be an algebraically closed field, and $f_0,\dots,f_m \in k[x_0,\dots,x_n]$ be homogeneous polynomials of the same degree. Denote by $I\subset k[x_0,\dots,x_m]$ the kernel of the homomorphism ...
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1answer
51 views

Decomposition of a monomial ideal

I have to find a primary decomposition of the following ideal and I proceeded in this way: $$(x^2z,x^2y^3,xt^2)=(x)\cap(t^2,x^2z,x^2y^3)=(x)\cap(t^2,x^2)\cap(t^2,z,z^2y^3)=(x)\cap(t^2,x^2)\cap(t^2,z,...
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0answers
25 views

$R$ integral domain, $P$ projective and injective module $\implies P=0$ or $R$ is field of fractions [duplicate]

I'm having difficulties proving the following: Let $R$ be an integral domain and $P$ a projective and injective $R$-module. Show that $P=0$ or $R=Q(R)$, where $Q(R)$ denotes the field of fractions of ...
3
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1answer
72 views

Irreducible elements for a commutative ring that is not an integral domain

Why does the definition of an irreducible element require us to be in an integral domain? Why can we not define an irreducible element exactly the same in a commutative ring that is not an integral ...
3
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3answers
106 views

Finitely generated projective modules over polynomial rings with integral coefficients

There is famous Quillen-Suslin theorem which states that every finitely generated projective module over a ring of polynomials $k[x_1,...,x_n]$, where $k$ is a field, is free. I have never carefully ...
2
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2answers
89 views

How do I find the ideal $I+J$ and quotient $R/(I+J)$?

This is a homework problem: Consider the polynomial ring $R=\mathbb Z_2[x_0,x_1,\dots,x_n]$. Let $I=\langle x_0x_1\cdots x_n\rangle$ and $J=\langle x_0+x_1,x_0+x_2,\dots,x_0+x_n\rangle$. Find $I+J$...
3
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0answers
47 views

Is quotient of open invariant subset open?

I am reading GIT book by Mumford. He needs special cases of the following conjecture several times. Conjecture Let $G$ be a reductive algebraic group acting on an irreducible affine scheme $X=Spec ...