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

learn more… | top users | synonyms (1)

0
votes
1answer
42 views

Computing a regular sequence of generators for an ideal

Let $R = \mathbb{C}[x_1,\ldots,x_n]$. Let $I$ be an ideal, and suppose we know a finite list of generators for $I$, say $I = \langle f_1,\ldots,f_k\rangle$. Is this information enough to compute a ...
1
vote
0answers
42 views

Finitely generated idempotent ideal must be generated by an idempotent [duplicate]

Let $A$ be a commutative but not necessarily unital ring. How can we show that a finitely-generated ideal $I$ of a ring $A$ satisfying $I=I^2$ is generated by an idempotent element?
0
votes
1answer
46 views

Noether normalization and surjectivity (revisited)

Let $Y$ be an affine variety of dimension $d$ inside the affine space $\mathbb{A}^n$. Then $A(Y) = k[x_1,\dots,x_n]/I_Y=:k[\bar{x}_1,\dots,\bar{x}_n]$. By the Noether normalization theorem, there ...
5
votes
1answer
83 views

(Co)homology of free symmetric algebra

Let $V$ be a (co)chain complex, and let $Sym(V)$ be the free differential graded-commutative algebra generated by $V$. Definition and examples below in case you don't know what I mean. Question: ...
0
votes
4answers
74 views

Maximal ideal in the ring of polynomials over $\mathbb Z$

Let $\mathbb Z[x]$ the ring of polynomials with integers coefficients in one variable and $I =\langle 5,x^2 + 2\rangle$, how can I prove that $I$ is maximal ideal. I tried first see that $5$ and ...
0
votes
1answer
37 views

Relation between faithfully flatness and map of $Spec$

I'm stuck on this exercise ( from Bosch ) : Let $\phi :R \to R' $ a flat ring morphism. Show that $\phi$ is faithfully flat if and only if the associated map $Spec(R') \to Spec(R)$ , ...
1
vote
1answer
28 views

Canonical homomorphisms $R_{\mathfrak{p}_i} \to R/\mathfrak{p}_i^n$ are isomorphisms when $R$ is artinian

I'm doing this exercise (from the book of Bosch): Let $R$ be an Artinian ring and let $\mathfrak{p}_1, \ldots \mathfrak{p}_n $ be its (pairwise different) prime ideals. Show that: a) The ...
0
votes
1answer
42 views

Common equivalent conditions for Cohen-Macaulayness of a ring

I know the fact that a local ring $(R,m)$ with $\dim(R)=d$ is Cohen-Macaulay (C-M) if and only if any one of the following holds: 1) $\operatorname{grade}(m)=\operatorname{height}(m)$ 2) ...
1
vote
1answer
51 views

Rings which are finitely generated and free over Cohen-Macaulay rings are also Cohen-Macaulay

Let $S$ be a Cohen-Macaulay (C-M) ring, and $R$ a ring containing $S$ such that as an $S$-module is finitely generated free. Could we deduce that $R$ is also C-M? I guess probably we could use ...
7
votes
1answer
164 views

Pseudo associated primes and short exact sequences

Let $A$ be a commutative ring, and $$0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow0$$ a short exact sequence of $A$-modules. The following inclusion relation is well-known: ...
4
votes
5answers
81 views

Show that $f^{-1}(\langle0\rangle)$ is not a maximal ideal of $\mathbb{Z}$.

Let $f\colon \mathbb{Z} \to \mathbb{Q}$ be a ring homomorphism. Show that $f^{-1}(\langle0\rangle)$ is not a maximal ideal of $\mathbb{Z}$.
0
votes
1answer
39 views

About second uniqueness primary decomposition theorem

I'm self-learning commutative algebra from Introduction To Commutative Algrebra of Atiyah and Macdonald and get frustrated about the second uniqueness primary decomposition theorem. I copy the theorem ...
0
votes
1answer
32 views

When a monomial ideal is primary

I know that a monomial ideal in $k[x_1, \ldots x_n]$ with $k$ a field is prime if and only if is of the following type $$I = (x_{i_1}, \ldots \ ,x_{i_k})$$ Is there a similar criterion to establish ...
0
votes
1answer
51 views

Height of finitely generated ideals in a catenary local ring

If $R$ is a noetherian local domain which is catenary, and $a_1,...,a_n$ are elements of the maximal ideal of $R$ with $\operatorname{height}(a_1,...,a_n)=n$, could we conclude that ...
1
vote
1answer
64 views

Primary decomposition of $I = (x^2, y^2, xy)$

I want to find a primary decomposition of the ideal $$ I = (x^2,y^2,xy) \subset k[x,y]$$ where $k$ is a field. How to proceed? Are there algorithms to find such decompositions? Where can I find ...
2
votes
1answer
89 views

Is the ring $ R = \{ f \in \mathbb{C}[x,y] \mid {\nabla f}(0,0) = (0,0) \} $ Noetherian?

Question: Is the ring $ R = \{ f \in \mathbb{C}[x,y] \mid {\nabla f}(0,0) = (0,0) \} $ Noetherian? I guess it isn’t Noetherian as I suspect that $$ (x y + y^{2}), \quad (x y + y^{2},x^{2} y + ...
1
vote
1answer
73 views

Finitely many prime ideals $\Rightarrow$ cartesian product of local rings

I'm stuck on this problem from Bosch, Algebraic geometry and commutative algebra: Let $R$ be a commutative ring containing only finitely many prime ideals and assume that a certain power of the ...
0
votes
1answer
44 views

Grade of non principal Prime ideals in Noetherian UFDs

I want to prove that in any Noetherian UFD the grade of every non-principal prime ideal is at least $2$. I say in a UFD $R$ each nonzero prime ideal contains a prime element. Since the given ...
0
votes
0answers
52 views

An example of regular local ring and its regular system of parameters

In the book of Eisenbud, "Commutative Algebra with a view toward Algebraic Geometry" it is quoted that "if $p$ is a prime integer, then $\mathbb Z_{(p)}[x_1,...,x_n]_{(p,x_1,...,x_n)}$ is a regular ...
0
votes
0answers
30 views

Criterion of separability of function on a curve.

Let $K$ be finitely generated extension of an algebraically closed field $k$ of transcendence degree $1$, $\operatorname{char }k =p$. Let $C(K)$ be set of discrete valuation rings $(\mathcal ...
-2
votes
1answer
45 views

Affine $K$-algebra is Hilbert ring!? [duplicate]

We know that when $F$ is a field, the ring $F[x_1,...,x_n]$ is a Hilbert ring, because the field $F$ is a Hilbert ring. My questions: Is any (non-trivial) affine algebra over an algebraically ...
1
vote
1answer
42 views

Associated primes and Noetherian condition

Let $A$ be a commutative ring, $M$ an $A$-module, and $N\subset M$ a submodule. Consider the following two sets: $$\Omega:=\{\mathfrak{p}\in\operatorname{Spec}A \ | \ \mathfrak{p}=(N:m) \ \mbox{for ...
1
vote
1answer
56 views

Ring extension and Jacobson rings

If $R\subseteq S$ are commutative rings, is it a fact that $R$ is a Jacobson ring if and only if $S$ is so? I guess the contraction of maximal and prime ideals of $S$ may be helpful in this ...
0
votes
1answer
50 views

Dimension of a semilocal Hilbert ring is zero

Is the Krull dimension of any commutative semilocal Hilbert ring equal to zero? I appreciate any help from anyone!
0
votes
0answers
18 views

Verification of an argument regarding the multiplication of polynomials.

Let $A[[x]]$ be the ring of formal power series $\sum_{n=0}^\infty{a_nx^n}$, where $A$ is a commutative ring. Prove that if $f\in A[[x]]$ is a unit, then $a_0$ is a unit in $A$. Is the following ...
3
votes
1answer
41 views

A question from Atiyah-Macdonald

Let $A$ be a commutative ring. I'm trying to prove that in $A[x]$, the Jacobson radical $\mathcal{J}$ is a subset of the nilradical $\mathcal{P}$. Let $a_0+a_1x+a_2x^2+\dots +a_nx^n\in\mathcal{J}$. ...
0
votes
1answer
43 views

Nullstellensatz non-valid for non-algebraically closed fields

I want an example (with details, please) showing that Nullstellensatz may be false over non-algebraically closed fields. Thanks in advance!
1
vote
2answers
59 views

intersection of non zero prime ideals of polynomial ring R[x] over integral domain R is zero

Let R be an integral domain. Then how to show that intersection of non zero prime ideals of R[x] is zero.
5
votes
3answers
121 views

Commutative ring with an ideal that contains all the nonunits

Is there an example of a commutative ring with an ideal that contains all the non-units? I was trying to think of some subring of $\mathbb Q$, but I couldn't get it to work.
1
vote
1answer
28 views

To show a certain integral closure is an order.

I have a Dedekind domain $R$ with field of fractions $K$ and a finite separable field extension $L$ of $K$. Let $S$ be the closure of $R$ in $L$. Is there a quick way to show that $S$ is finitely ...
0
votes
2answers
50 views

“Finitely generated as an $R$-module”

Please could somebody explain to me what it means for something to be finitely generated as an $R$-module? I can't seem to find a definition anywhere! Thanks!
0
votes
1answer
34 views

A local subring of $F[[x]]$?

Suppose that $F$ is a field and $R=F⊕x^2F[[x]]$, where $F[[x]]$ is the ring of power series in one indeterminate $x$ with coefficients in $F$. I guess that $R$ is a local ring with the maximal ...
0
votes
0answers
30 views

Relation between the initial ideal and radical

Let $I$ be an ideal of the polynomial ring $S$. Show that ${\rm In}(\sqrt I)\subseteq\sqrt{{\rm In}(I)}$, where by ${\rm In}(I)$ we denote the ideal of initial forms of I, In(I) = (In(f) : f $\in$ I). ...
1
vote
1answer
80 views

$\mathbb C[x_1,\ldots,x_n]/I=\mathbb C\times\cdots\times\mathbb C$.

Let $A=\mathbb C[x_1,\ldots,x_n]/I$ and for every $y\neq 0$, we have $y^2\neq 0$ and $\dim A=0$. I would like to prove that $A=\mathbb C\times\cdots\times\mathbb C$. Attempt of a solution $\dim ...
0
votes
1answer
40 views

Intersection of $max(R)$ with a closed subset in $Spec(R)$

Let $R$ be a commutative ring with unity and $E$ be a nonvoid closed subset of $Spec(R)$. If $U$ is an open subset of $Spec(R)$ with $E∩Max(R)⊆U$, where $Max(R)$ is the set of maximal ideals of $R$, ...
2
votes
1answer
63 views

Jacobson radical of a ring finitely generated over $\mathbb Z$

If a commutative ring $R$ with $1$ is finitely generated over $\mathbb Z$ could one deduce that the Jacobson radical of $R$ is nilpotent? I am aware of the well-known fact that when $R$ is ...
6
votes
0answers
98 views

If $a,b$ is an $R$-sequence, then $(ax-b)$ is prime [duplicate]

If $R$ is an integral domain, $a, b\in R, a\neq 0$ and $\bar b$ is not a zero divisor in $R/(a)$. I'm trying to prove $(ax-b)\in R[x]$ is prime. This question seems easy but I couldn't prove it, ...
0
votes
3answers
86 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 ...
1
vote
0answers
28 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
91 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
393 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 $ab\in I$, ...
1
vote
1answer
39 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
54 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 ...
0
votes
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
64 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
53 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
vote
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
49 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)}} ...