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

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0
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3answers
84 views

This ideal is not maximal [duplicate]

I'm trying to prove this ideal: $$(x^2+y^2+z^2+x+y+z,x^5+y^5+z^5+2(x+y+z),x^7+y^7+z^7+3(x+y+z))\subset \mathbb C[x,y,z]$$ Can't be maximal. In order to do so, I'm using the Nullstellensatz ...
0
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0answers
13 views

Elimination order on polynomial ring

Having the polynomial ring $K[x,y]$ is there an unique elimination order for $x$?
1
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0answers
27 views

An inverse limit of a certain inverse system

Let $∆$ be a directed set and $(N_i,f_{ji})_{i∈∆}$ be an inverse system of $R$-modules. Fix $α \in∆$ and consider $(M_i,g_{ji})_{i\in∆}$ as follows: $M_i=N_i$ for $i≥α$, $M_i=0$ for $i<α$, and ...
4
votes
1answer
90 views

$\dim \mathbb K[x,y,z]/(xy,xz,yz)$

If $\mathbb K$ is a field I would like to find $$\dim \mathbb K[x,y,z]/(xy,xz,yz)$$ I'm starting to study the concept of dimension of rings and I don't know the basic tools and techniques to discover ...
3
votes
4answers
392 views

This ideal is prime

I'm trying to prove this ideal $$I=(x^2+y^2+x,x+y+xy)\subset \mathbb C[x,y]$$ is prime. I supposed that $I$ is prime and I'm using the classical method to prove $I$ is prime: If $ab\in I$, ...
0
votes
1answer
37 views

Set maps given by a polynomial & Yoneda Lemma

This Exercise 4.1. from the book Algebraic Geometry I, by Gortz. Problem Let $R$ be a ring, and for every $R$-algebra $A$ let $\alpha_A:A\rightarrow A$ be a map of sets such that for every ...
0
votes
1answer
53 views

Ideal equals the whole ring

Show that in the polynomial ring $S=K[x_1, ..., x_n]$, having an ideal $I = (g_1, ..., g_m)$ and ${g_1, ..., g_m}$ a Groebner bases of $I$, then $I = S$ if and only if one of the $g_i$ is a nonzero ...
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0answers
48 views

Groebner bases for sum of ideals

$S = K[x_1, \ldots x_n]$. Let $I,J \subset S$ be ideals and $<$ a monomial order on $S$. Let $G, G'$ be Groebner bases of $I$, respectively $J$ with respect to $<$. Prove that if ...
2
votes
1answer
62 views

Integral domain with a finitely generated non-zero injective module is a field

Suppose that $R$ is a integral domain. Suppose that there exists a non-zero finitely generated injective module $M$. How can I prove that $R$ is field?
2
votes
1answer
52 views

Quotients of $p$-adic completion

Let $R$ be a commutative ring and $p \in R$. Consider the $p$-adic completion $\widehat{R} := \varprojlim_{n} \, R/p^n$. When do we have $\widehat{R}/p^n \widehat{R} \cong R/p^n R$? For fixed $n$ ...
1
vote
0answers
142 views

Generalization of Chinese Remainder Theorem to infinite ideals

I'm looking for any (obviously weaker) generalization of this famous theorem in the special case that the family of ideals is not finite.
1
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0answers
34 views

Noetherian rings/Hilbert's Basis Theorem

So I'm studying the proof of Hilbert's Basis Theorem - we've shown that $λ(I)$ is an ideal of $R$ and and then it says "Since R is Noetherian, we have $λ(I) = \sum\limits_{i=1}^k s_iR$ for some $s_1, ...
0
votes
1answer
66 views

$2$-dimensional Noetherian integrally closed domains are Cohen-Macaulay

Any 1-dimensional Noetherian domain is Cohen-Macaulay (C-M). For the $2$-dimensional case, a condition of being integrally closed is necessary to be added for a Noetherian domain to be C-M, which ...
0
votes
2answers
48 views

$A = \bigcap_{\mathfrak{p} \in \text{Spec(A)}} A_{\mathfrak{p}} = \bigcap_{\mathfrak{m} \in \text{MaxSpec(A)}} A_{\mathfrak{m}}$

I'm doing this exercise. Let $A$ be an integral domain, then prove that $$A = \bigcap_{\mathfrak{p} \in \text{Spec(A)}} A_{\mathfrak{p}} = \bigcap_{\mathfrak{m} \in \text{MaxSpec(A)}} ...
0
votes
1answer
29 views

Non-Noetherian ring $R$ with Spec($R$) a Noetherian Scheme

In looking at the examples of Non-Noetherian rings I knew/found I wasn't able to find one where I could conclude that Spec($R$) was a Noetherian scheme (not just merely a Noetherian topological ...
0
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0answers
33 views

Localization of Coordinate Ring isomorphic to ring of local regular functions

There is a very standard fact that I am having a hard time understanding. The claim is that if we have an affine variety $Y$ then the localization of the coordinate ring by the maximal ideal ...
1
vote
1answer
61 views

Free modules over local Artin rings

Let $A$ be a local Artin ring, $B$ be a $A$-flat algebra and $M$ a finitely generated $B$-module and flat $A$-module. Let $k$ be the residue field of $A$ i.e., $A/m$ where $m$ is the maximal ideal of ...
0
votes
0answers
62 views

An $n$-generated ideal of grade $n$ can be generated by an $R$-sequence in any order

It is known to me that if $I$ is an $n$-generated ideal of a commutative ring $R$ with $\operatorname{grade}(I)=n$, then it is generated by an (ordered) $R$-sequence in $I$ of length $n$. I have a ...
0
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0answers
23 views

What is the purpose of the generalised definition of a cluster algebra?

The seeds of a cluster algebra are normally of the form $(\textbf{x},B)$ where $\textbf{x}$ is a cluster and $B$ is a skew-symmetrizable matrix. However then I have come across a more general ...
0
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1answer
57 views

A question on localization of fractional ideals

I have a domain $A$ with field of fractions $K$ and a non-zero fractional $A$-ideal $I$. Let $I'$ be the fractional ideal $\{a\in K\mid aI\subset A\}$. I assume that $II'\subset \mathfrak p$ for ...
4
votes
1answer
46 views

Eisenstein integers and $\mathbb{Z}C_3$

The Eisenstein Integers $a+b\omega$ with norm $N(x)=a^2-ab+b^2$ form a commutative ring, as does the group ring $\mathbb{Z}C_3=\{\sum_{g\in C_3} a_g g \mid g \in C_3, a_g \in \mathbb{Z}\}$. ...
0
votes
2answers
71 views

Grade of an ideal in a Noetherian ring

I want to prove that if $R$ is a Noetherian integral domain and $I$ is a nonzero ideal of $R$, then $I^{-1}=R$ if and only if $\operatorname{grade}(I)≥2$. For the "only if" part, I say ...
3
votes
2answers
103 views

Exterior power “commutes” with direct sum

I know that for vector spaces $V, W$ over a field $K$, we have the following identity : $$ \bigoplus_{k=0}^n \left[ \Lambda^k(V) \otimes_K \Lambda^{n-k}(W) \right] \simeq \Lambda^n(V \oplus W) $$ ...
6
votes
2answers
65 views

Why is it Artinian?

The following is a part of the section entitled Samuel functions in the book Commutative Ring Theory by Hideyuki Matsumura: Let $A$ be a Noetherian semilocal ring, and $\mathfrak{m}$ the ...
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0answers
29 views

Suggest a good book or reference on graded modules over polynomial rings

I am looking for reference books or papers on graded modules over the polynomial ring $k[x_0, \ldots, x_n]$. Any good commutative algebra text like Eisenbud's Commutative Algebra already contains a ...
3
votes
1answer
58 views

Direct product of Cohen-Macaulay rings/Eisenbud, Exercise 18.6

Somehow I believe (or doubt (!)) that direct product of two Cohen-Macaulay (C-M) rings may not be C-M. Can anybody give me an example verifying this? I would be grateful to him/her.
1
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0answers
28 views

Prove that $B_{q}$ is flat over $B_{p}$

I'm doing this exercise in "Introduction to Commutative Algebra" of Atiyah and get confused by the hint in this book. Here is the exercise: Let $f: A \rightarrow B$ be a flat homomorphism of ...
0
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0answers
51 views

Sufficient conditions for quotient ring to be Cohen-Macaulay

We know that every Noetherian integral domain with (Krull) dimension $1$ is Cohen-Macaulay (CM). In a commutative algebra text the author have presented the following problem: "Let $(R,m)$ be a CM ...
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votes
1answer
113 views

An ideal which is not maximal in $\mathbb{C}[x,y,z]$

Show that $$J=(x^2+y^2+z^2+x+y+z, x^5+y^5+z^5+2(x+y+z), x^7+y^7+z^7+3(x+y+z))$$ is not the maximal ideal $m=(x,y,z)$ in $\mathbb{C}[x,y,z]$.
1
vote
1answer
59 views

Grade of an ideal equal to that of all its associated primes?

By a well-known theorem in commutative algebra, if $I$ is an ideal of a commutative ring $R$ then $\operatorname{grade}(I)$ is the minimum of $\operatorname{grade}(P_i)$, where $P_i$ are the ...
4
votes
3answers
174 views

Question about fields and quotients of polynomial rings

I don't see how to solve the following problem: Let $R$ be a commutative and unitary ring. If there exists a monic polynomial $f(x) \in R[x]$ so that $R[x]/(f(x))$ is a field, show that $R$ is a ...
0
votes
1answer
61 views

Krull dimension of $k[x,y,z,t]/(xy-z^3,z^5,x^2t^3+y^2)$. [closed]

I need help to solve this exercise. If anyone can help, thanks in advance! Let $k$ a field and $R=k[x,y,z,t]/(xy-z^3,z^5,x^2t^3+y^2)$. Find the Krull dimension of $R$.
2
votes
1answer
26 views

Irredundant intersection of submodules

Let $A$ be a commutative ring, $M$ an $A$-module, and $N_\alpha\subset M$ a family of submodules. Consider the intersection $$\bigcap_\alpha N_\alpha.$$ We say that the intersection is irredundant if ...
0
votes
1answer
24 views

$Hom_{A-Alg}(Sym(M),B)\cong Hom_{A-mod}(M,B)$

Let A be an ring. M be an A-module. Let $Sym (M)$ be the Symmetric algebra of M over A. Let B be an A-algebra. Why is $Hom_{A-Alg}(Sym(M),B)\cong Hom_{A-mod}(M,B)$ One way is clear - If we have a ...
0
votes
0answers
44 views

prove $\dim\mathbb{Z}[X_1,X_2]=3$ from first principles

Since $\dim R[X] =\dim R+1$ for any Noetherian ring $R$, the ring $\mathbb{Z}[X_1,X_2]$ must have dimension 3. But how can this be proved 'from first principles', i.e. without using any big theorems ...
2
votes
1answer
25 views

Can the submodule generated by action of nilradical be equal to whole module

Let $A$ be a commutative ring with unity, $N$ be the nilradical of $A$, $M$ be an $A$-module. Is it always true that $NM$ is a proper submodule of $M$? If $M$ is finitely generated then by Nakayamma ...
2
votes
2answers
88 views

Two nonassociated functions defining the same hypersurface?

Let $X\subseteq\mathbb P^n$ be a complex, irreducible projective variety. Let $R$ be the projective coordinate ring of $X$, i.e. $R=\mathbb C[x_0,\ldots,x_n]/I$ for some homogeneous prime ideal $I$. ...
3
votes
2answers
80 views

Normal if and only if is UFD

If we consider $f \in \mathbb{C}[x,y]$ an irreducible polynomial, then it is true that the domain $ \mathbb{C}[x,y]/(f)$ is normal iff it is UFD? I think this is false. I was trying to prove ...
4
votes
1answer
41 views

Local ring with intersection of powers of its principal maximal ideal zero

In an algebra test the following problem was presented: Any commutative local ring $(R,m)$ with $m$ principal so that $⋂_{i≥0}m^i =0$ is Noetherian and each nonzero ideal of $R$ is a power of ...
1
vote
2answers
48 views

UFD yields height of certain primes at most $1$

Let $R$ be a unique factorization domain. If $P$ is a prime ideal minimal over a principal ideal, is it true that height of $P$ is at most $1$? In case $R$ is Noetherian the result follows due ...
0
votes
1answer
51 views

Powers generate monomials

What is a reference in the literature for the following fact? Let $A$ be a commutative $\mathbb{Q}$-algebra. Then every monomial in $A$ of degree $n$ may be written as a linear combination of $n$th ...
3
votes
0answers
54 views

Topological characterisations of freeness and separability?

According to wikipedia, a spectral space is defined to be homeomorphic to the spectrum of some commutative ring. They form a category $Spec$ where we take morphisms to be those whose preimage of open ...
0
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0answers
57 views

Different generators of (x,y) in k[x,y] give rise to automorphism.

I am stuck with the following algebra problem: Let $f,g\in k[x,y]$ be polynomials which generate $(x,y)$ (as an ideal). Consider the homomorphism $\phi:k[x,y]\to k[x,y]$ which is identity on $k$, and ...
-1
votes
1answer
49 views

Open set in the spectrum of a ring?

Consider $Spec(K[X])$ where $K$ is an algebraically closed field. Is $0$ open in the Zariski topology on spectrum? Does the spectrum have points which are neither open nor closed?
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0answers
23 views

When is the number of prime ideals lying over a given one finite?

Let $R \to S$ be a ring extension and let $\mathfrak{p}$ be a prime ideal of $R$. Question: Under what conditions on the extension is the number $N(\mathfrak{p})$ of prime ideals lying over ...
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votes
1answer
58 views

How to understand the regular sequence of a module

If we have a regular sequence $a_1,\dots, a_r$ in a ring $A$, I think it means the subschemes $A/(a_1,\dots,a_i)$ cut out step by step are all equi-dimensional. (when $A$ is affine coordinate ring, by ...
1
vote
1answer
56 views

Is the integral closure of local domain a local ring?

Suppose $A$ is a local domain, with field of fractions $K$, let $A'$ be the integral closure of $A$ in $K$, is $A'$ a local ring?
2
votes
2answers
106 views

Examples of Noetherian local rings which are not Gorenstein

Can anyone give me an example of a Noetherian local ring which is not a Gorenstein ring?
3
votes
3answers
62 views

Injective dimension of $\mathbb Z_n$ as a $\mathbb Z$-module

What is the injective dimension of $\mathbb Z_n$ as a $\mathbb Z$-module? Can one use the well-known fact that $id(M)$ is less than or equal to $i$ iff $Ext^{i+1}(N,M)=0$ for all $N$? Thanks in ...
0
votes
0answers
39 views

Equivalent definitions of Jacobson rings

We say that a ring $R$ is a Jacobson ring if $$ J(R/I)=\operatorname{nil}(R/I) $$ for every proper ideal $I$ of $R$, where $ J(R)=\bigcap\{M:M \text{ maximal ideal}\}. $ Then it also says, ...