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

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2
votes
2answers
84 views

The unit group of $\mathbb{Q}[x, y]/(x^2+y^2+1)$

During some calculations, I encountered with the problem of calculating the unit group of the $\mathbb{Q}$-algebra $\mathbb{Q}[x, y]/(x^2+y^2+1)$. I believe it is the unit group of the field of ...
0
votes
1answer
49 views

Primary Ideal and Associated Primes [duplicate]

I'm trying to understand the proof of the following statement: If $R$ is Noetherian, then an ideal $Q$ is $P$-primary for a prime $P$ $\Leftrightarrow$ $Ass(R/Q)=\lbrace P \rbrace$. I can show the ...
3
votes
0answers
36 views

Equivalence of definitions for completion

For the settings on my question, take Atiyah's chapter on completions. Basically we have two definitions of completness (Atiyah's sense, the canonical map $\phi:M\rightarrow \widehat{M}$ is an ...
2
votes
1answer
30 views

An example of a (necessarily non-Noetherian) ring $R$ such that $\dim R[T]>\dim R+1$

What is an example of a non-Noetherian ring $R$ such that the Krull dimension of $R[T]$ is greater than dim$R+1$?
-1
votes
1answer
53 views

How to prove that the set of maximal elements of a set of prime ideals is finite

Let $A$ be a subset of ${\rm Spec}(R)$ with $R$ noetherian Are there any techniques to prove that ${\rm max}(A)$ (ie the set of maximal elements of $A$) is finite? I'm looking for equivalent ...
4
votes
0answers
44 views

Prime ideals contained in the union of almost all prime ideals

I am reading the proof of the long exact sequence involving $S$-class groups and $S$-units in Neukirch Algebraic Number Theory, Chapter I, Prop. 11.6, which states the following canonical sequence is ...
-1
votes
0answers
35 views

Stable Filtrations and Completions

We know that two stable filtrations define the same topology on a module, hence isomorphic completions. I want to show that this is true when we use the definition of inverse systems. This is, if ...
0
votes
0answers
102 views

Unique factorization in prime ideals in a local ring

Let $R$ be a local commutative domain with maximal ideal $M$. Assume that every ideal of $R$ is a product of prime ideals in a unique way. I want to show that the only non-zero prime ideal of $R$ ...
2
votes
0answers
31 views

Does the relation $\mid^*$ have any interesting applications for understanding the structure of commutative rings that aren't integral domains?

There is a binary relation $\mid^*$ defined on any commutative ring as follows: $a \mid^* b$ iff $ak=b$ for some $k \in R$ that is not a zero divisor. This is always transitive, and it is reflexive ...
5
votes
1answer
50 views

Proving a ring is Noetherian when all maximal ideals are principal generated by idempotents

Let $R$ be a commutative ring with unity such that all maximal ideals are of the form $(r)$ where $r\in R$ and $r^2=r$. I wish to show that $R$ is Noetherian. I know that if all prime (or ...
0
votes
1answer
40 views

Show that $f^*:Spec(B)\rightarrow Spec(A)$ is a closed mapping.

Let $f:A \rightarrow B$ be an integral homomorphism of rings. Show that $f^*:Spec(B)\rightarrow Spec(A)$ is a closed mapping. My Try: So, $B$ is integral over $f(A)$. $f^*$ is given by $b\longmapsto ...
6
votes
1answer
74 views

How to imagine the difference between the following schemes?

Consider $A=\operatorname{Spec} k[x]_{(x)}[t]$ and $B=\operatorname{Spec} k[x,t]_{(x)}$ for a field $k$ (Vakil, note 11.3.8). For me, both are infinitesimal neighborhoods of an affine line - the ...
2
votes
1answer
71 views

Finding the kernel of $\alpha: K[X,Y,Z]^{3}\rightarrow \langle X,Y,Z\rangle$, $(f,g,h)\mapsto Xf+Yg+Zh$.

I am trying to do exercise $2.3$ of Reid's "Undergraduate Commutative Algebra": Let $A=K[X,Y,Z]$ where $K$ is a field, and $m=\langle X,Y,Z\rangle$. I have to show that the kernel of the ...
2
votes
1answer
46 views

Is $x \in X$ isolated iff the ideal $M_x := \{f \in C(X, \mathbb R) :f(x)=0\}$ of $C(X, \mathbb R)$ is principal ?

Let $(X,d)$ be a metric space , then is $x \in X$ isolated iff the ideal $M_x := \{f \in C(X, \mathbb R) :f(x)=0\}$ of $C(X, \mathbb R)$ is principal ? Do we need completeness of $X$ ?
1
vote
1answer
21 views

If $(X,d)$ is a finite metric space , then is every prime ideal of $C(X, \mathbb R)$ maximal?

If $(X,d)$ is a finite metric space , then is every prime ideal of $C(X, \mathbb R)$ maximal ? The thing is , since $X$ is finite , so it is compact , so ideal $M$ is maximal iff it is of the form ...
1
vote
1answer
35 views

Does the equation $\operatorname{Ass}M=\operatorname{Ass}E(M)$ hold for non-finitely generated modules $M$?

Does the equation $\operatorname{Ass}M=\operatorname{Ass}E(M)$ hold for non-finitely generated modules $M$? Here $E(M)$ is the injective envelope of $M$ and $\operatorname{Ass}$ denotes the set ...
5
votes
2answers
48 views

Dimension of Tensor Product for Flat Extensions

Suppose that $A,B,$ and $C$ are commutative unital rings, $A\to B$ is flat, and $A\to C$ is any map. I am trying to determine whether $$ \dim B\otimes_AC=\dim B+\dim C-\dim A $$ Any counterexamples ...
1
vote
1answer
47 views

Determine whether $(\mathbb{R}[x,y]/(y^2-x^2-x^3))_{(x,y)}$ is a discrete valuation ring.

Geometrically, the curve $y^2-x^2-x^3=0$ is singular at the origin in the real plane. Thus the ring should not be a dvr. I am thinking to show that it is not a dvr, it is equivalent to show that it ...
0
votes
1answer
34 views

Polynomial ring with integral coefficients is integral

Let $B$ be a ring and $A\subset B$ a subring. Assume that $B$ is integral over $A$. I have to prove that $B[X]$ is integral over $A[X]$. I tried writing down an integral relation for $f(X)\in B[X]$ ...
1
vote
2answers
81 views

If local rings are Noetherian, is scheme locally-Noetherian?

If all the local rings of a scheme are Noetherian, is the scheme locally-Noetherian?
0
votes
1answer
18 views

A question concerning to show that $V(I)$ is open if $I$ is radical ideal

Let $I:=(f_1 ,...,f_k)$ be a finitely generated ideal of $C(X,\mathbb R)$ such that $\mathrm{rad}(I)=I$, $f:=\sqrt{\sum_{m=1}^k |f_m|}=\sum _{m=1}^k g_mf_m$ where $g_i $'s are real valued ...
0
votes
0answers
51 views

Kernel identity of the canonical homomorphism

Let $A$ be a commutative complete Noetherian ring with unit for the $I$-adic topology, where $I$ is an ideal of $A$. Suppose that $M_0$ is finitely generated over $A$. Let $(M_n)_{n\geq 0}$ be ...
4
votes
0answers
112 views

Determine all discrete valuations on $\mathbb{C}(x)$.

To clarify, for a field $K$, a valuation $v$ on $K$ is a map $v:K^{\times}\to G$ for $G$ an ordered group (written additively) such that for any $a,b\in K^{\times}$: 1) $v(ab)=v(a)+v(b)$; 2) ...
0
votes
0answers
31 views

What is interesting (useful) about Multiplicity?

Multiplicity is defined at 4.1.5; Bruns_Herzog. People say it is an important invariant. I don't know what idea is behind this definition and What is interesting/useful about it. what important ...
3
votes
1answer
43 views

Pushforward of a Cohen-Macaulay sheaf

Let $f: X \to Y$ be a finite surjective morphism of quasi-projective varieties. Let $X$ be Cohen-Macaulay and let $Y$ be smooth. Now let $\mathcal{F}$ be a coherent sheaf on $X$. Then $f_* ...
3
votes
0answers
58 views

Theorem 1 in chapter II.4 of Mumford's Red Book

while reading Mumford's wonderful Red Book, I arrived to a Theorem where I don't understand the proof. So Theorem 1 in chapter II.4 says Let $X_0$ be a prescheme over $k_0$, let $X= X_0 ...
0
votes
1answer
42 views

Question about the algebraic definition of tangent space

Let $V\subset \mathbb{A}^n(K)$ be an affine algebraic set and let $K[V]$ be its coordinate ring. For $a\in V$, let $T_aV$ be the tangent space of $V$ at $a$ and $P=I(a)$ be the maximal ideal in $K[V]$ ...
3
votes
1answer
63 views

Grobner basis and subsets

Let $A$ be a subset and $I$ an ideal of polynomial ring $R=k[x_1,x_2,...,x_n]$. Is there any algorithm for deciding when $A\subseteq I$?
2
votes
1answer
45 views

When is the Frobenius the identity?

If $f$ is an irreducible polynomial of degree $n$ over $\mathbb{F}_{p}$, then $\mathbb{F}_{p}[x]/(f)$ is the finite field $\mathbb{F}_{p^{n}}$ and the map $a \mapsto a^{p}$ is the Frobenius ...
0
votes
0answers
49 views

Gluing construction of the projective space scheme.

When constructing the projective space scheme $\mathbb{P}_R^n$ for a ring $R$, we may take the subrings $$ A_i = R\left[\tfrac{X_0}{X_i}, \ldots, \widehat{\tfrac{X_i}{X_i}}, \ldots, ...
2
votes
2answers
150 views

Proof of Proposition 2.4 in Atiyah-MacDonald [closed]

I'm struggling here with the proof. To be honnest i need a really concrete explanation because i have been on it for a long time and i can not find it nowhere else. Please can anyone help me with ...
1
vote
1answer
92 views

Analysis of the ideals of $C[0,1]$

For every ideal $I$ of $C[0,1]$ , define $Z(I):=\{x \in [0,1] :f(x)=0 , \forall f \in I\}$ and for every $A \subseteq [0,1]$ , let $I(A):=\{f \in C[0,1] : f(x)=0 , \forall x \in A\}$ . Then ...
7
votes
0answers
110 views

Check whether a polynomial ideal is prime in the power series ring

I would like to know whether the ideal $I = \langle y^{2}(y^{2}-x^{2}) + w^{7}, y^{2}(y^{4}-x^{4}) + z^{7}\rangle$ is prime in $\mathbb{C}[[x,y,z,w]]$, the ring of formal power series in the ...
5
votes
3answers
119 views

Properties characterized by a vanishing Ext or Tor module

While reading Weibel's "An introduction to homological algebra'', I've noticed that many properties of a module are characterized by the vanishing of some Tor or Ext. Fix a (commutative) ring $R$ and ...
1
vote
1answer
46 views

Relation between the inverse of finitely generated ideals and the inverse of their powers.

Let $D$ be an integral domain, $K$ its field of fractions, and $J_1,...,J_n$ are ideals of $D$ such that $(\sum_{i=1}^{n} J_i)^{-1}=D$. How can we prove that this implies $(\sum_{i=1}^{n} ...
0
votes
0answers
46 views

$\mathrm{Ext}^i(-,A/\mathfrak{m})$ in $(A,\mathfrak{m})$ noetherian regular local ring

Dealing with $\mathrm{Ext}^i(\mathcal{F},k(x))$ on a smooth variety over a field $k$, with $\mathcal{F}$ coherent and $k(x)$ skyscraper sheaf of a closed point I foundin a proof that for $i=2,3$ (and ...
4
votes
2answers
36 views

Easy proof from Atiyah-McDonald on module homomorphisms

Let $v \in \text{Hom}(M,M')$ Then if the induced hom $\bar{v} : \text{Hom}(M',N) \to \text{Hom}(M,N)$ given by $f \mapsto f \circ v$ is injective for all $N$, then $v$ is surjective. Attempted proof ...
3
votes
0answers
32 views

Extension of Integral Domains

Let $S\subset R$ be an extension of integral domains. If the ideal $(S:R)=\{s\in S\mid sR\subseteq S\}$ is finitely generated, show that $R$ is integral over $S$. My first attempt was to show ...
2
votes
1answer
36 views

Closed and open subsets of Spec(R)

Suppose we are given a commutative ring with unit $R$ and an ideal $I \subseteq R$. I wondered if then the following is true: $$ V(I) \text{ is open in Spec}(R) \Leftrightarrow I = I^2. $$ I know that ...
0
votes
1answer
17 views

Characterisation of DVR via invertible ideals

I am reading these notes. In terms of the proof of proposition 9.17, I understand the writer used proposition 9.5 to show this. However, to use proposition 9.5, we should check before if our local ...
1
vote
1answer
44 views

On the Krull Dimension of Quotient Rings

Let $I\subset J \subset R$ be ideals of $R$. How can we show that $\dim(R/J) \leq \dim(R/I)$? So far I've shown that the heights meet $ht(I)\leq ht(J)$, which is fairly straight forward, but I am ...
0
votes
1answer
45 views

Prove or Disprove: Finitely generated Artinian module is Noetherian.

I think it is true and I am trying to prove it. I am considering reducing the case to Artinian rings. Say $M$ is finitely generated Artinian $R$-module. Then $R/Ann(M)$ is an Artinian $R$-module. Thus ...
2
votes
1answer
89 views

Atiyah-MacDonald, Exercise 5.4

I was having some trouble with the following exercise from Atiyah-MacDonald. Let $A$ be a subring of $B$ such that $B$ is integral over $A$. Let $\mathfrak{n}$ be a maximal ideal of $B$ and let ...
1
vote
1answer
36 views

Nontrivial map between finite modules over local noetherian ring

I have just read that over a noetherian local ring $(A,\mathfrak{m})$, there is always a nontrivial map between two finitely generated modules such that their support is exatly $\mathfrak{m}$. The ...
4
votes
1answer
52 views

Least rational prime which is composite in $\mathbb{Z}[\alpha]$?

Sébastien Palcoux asked if there was some irrational algebraic $\alpha$ such that all rational primes are primes in $\mathbb{Z}[\alpha].$ MooS answered that there are no such $\alpha.$ This leads to a ...
1
vote
1answer
46 views

Certain products of mostly diagonal matrices are nonzero

Let $F$ be the field ${\mathbb Q}(x,y)$ and let $n>0$ be an integer. Consider the following two matrices in $M_2(F)$ : $$D=\left(\begin{array}{cc} x & 0 \\ 0 & y\end{array}\right), ...
2
votes
0answers
60 views
-2
votes
1answer
83 views

Localization of Ring

I am trying to understand the justification for the last sentence of this theorem (http://stacks.math.columbia.edu/tag/00JA). I do not understand how they can say, "each $Ae_i$ is a ring". It is not a ...
2
votes
1answer
55 views

Module in which every submodule containing the radical is an intersection of maximal submodules.

$\newcommand{\Rad}{\operatorname{Rad}}$Let $M$ be a left $R$-module with $\Rad(M)\neq 0$ where $\Rad(M)$ is defined as the intersection of all maximal submodules of $M$. Thus, if $K = \bigcap_{i\in I} ...
1
vote
1answer
60 views

Question about the polynomial grade of a finitely generated ideal

Let $R$ be a commutative ring (not necessary Noetherian), $Q$ its total ring of fractions, and $I$ a finitely generated ideal of $R$ such that $\forall$ $a \in I$ we have $(a:_R I) = a$. My question ...