1
vote
1answer
42 views

For a discrete valuation ring to be a PID, must it have an element of valuation 1?

When is a discrete valuation ring a PID? Must it have an element of valuation 1 or is this not necessary?
0
votes
1answer
18 views

Bass numbers of a minimax module

Let $R$ be a commutative Noetherian ring, and $M$ be a minimax $R$-module. Are the Bass numbers of $M$ are finite? (An $R$-module $M$ is called minimax, if there is a finite submodule $N$ of $M$, such ...
0
votes
2answers
36 views

Modules with finite support in $\mathrm{Max}(R)$

Let $R$ be a commutative Noetherian ring, and $M$ be an $R$-module. Is the following statement true? If $\mathrm{Supp}_R(M)$ (support of $M$) is a finite subset of $\mathrm{Max}(R)$ (the set of all ...
0
votes
1answer
26 views

Example of commutative algebra over integers where there exists $x$ such that $x = y^2$ for several $y$'s

Is there a commutative algebra over integers such that there exists $x$ with $x = y^2$ for several $y$'s? Also, is there a commutative algebra over integers such that for every $k \in \mathbb{N}$, ...
0
votes
0answers
32 views

Exercise on localization as a colimit

I am doing the following exercise: Suppose $S$ is a multiplicative set of $A$, an integral domain, and interpret $S^{-1}A = \varinjlim \dfrac{1}{s}A$, where the limit is over $s \in S$ and in the ...
3
votes
0answers
34 views

Resultants of two polynomials over a ring

Let $k$ be a field $f,g\in k[x,y]$ be two polynomials. The resultant $R\in k[x]$ is a polynomial function of the coefficients of $f$ and $g$, such that $f$ and $g$ gave a common zero (in an extension) ...
1
vote
1answer
21 views

Rational group algebras and maximal orders

Let $G$ be a finite group, and $\mathbb{Q}[G]$ be the rational group algebra. Then the group ring $\mathbb{Z}[G]$ is an order in $\mathbb{Q}[G]$, but is not in general a maximal order. What can we ...
3
votes
1answer
36 views

Injective hull of $\mathbb{ Z}_n$ [duplicate]

What is the injective hull of $\mathbb Z_n$? I know that in case $n=p$ is prime, the injective hull would be isomorphic to $\mathbb Z_{p^∞}$, but in general case, I have no idea. Can anyone be of ...
0
votes
1answer
41 views

Isomorphism between $R$ and its dual space

Let $R$ be a finite dimensional algebra over a field $K$. If $f$ is an $R$-module monomorphism from $R$ to the dual $K$-space $\operatorname{Hom}_K(R,K)$ why it is onto? Thanks!
3
votes
0answers
92 views

Ring of rational power series

Let $A$ be any commutative ring with 1. A power series $f\in A[[t]]$ is called rational if we can find a $g\in A[t]$ such that $fg\in A[t]$. It is clear that the set of rational power series forms a ...
2
votes
1answer
96 views

Help in this notation in Fulton's Algebraic Curves book

I'm reading Fulton's Algebraic Curves book, I'm stuck in the following proposition (page 105): In fact, what I didn't understand is the following notation in the proof of this proposition: Why ...
0
votes
1answer
52 views

Proof that presheaf is a sheaf for Spec

Atiyah Macdonald define presheaf (chapter 3, exercise 23) on the base of $Spec(A)$, where $A$ is commutative ring with $1$, as follows $$ \mathfrak{F}(X_f) = A_f, $$ where $X_f$ is a basic open set ...
3
votes
1answer
112 views

Endomorphisms of the maximal ideal of a local ring

Let $R$ be a commutative local ring with maximal ideal $\mathfrak{m}$. Is it true in general that $\text{Hom}_R(\mathfrak{m},\mathfrak{m})\cong \text{Hom}_R(\mathfrak{m}, R)$? What if the Krull ...
2
votes
1answer
44 views

Von Neumann regular but not self-injective ring

I want an example of a von Neumann regular ring which is not self-injective. My thanks go to anybody answering.
2
votes
1answer
40 views

A statement related to Hilbert Nullstelensatz

By Hilbert Nullstelensatz we know that for any field $k$, every maximal ideal of $k[x_1, ..., x_n]$ has residue field a finite extension of $k$. I also did an exercise which goes: any integral domain ...
2
votes
1answer
32 views

Example of a ring which is not CM at all its prime ideals

A commutative ring $A$ is said to be CM at a maximal ideal $\mathfrak{m}$ if and only if $Depth(A_{\mathfrak{m}})=Krull(A_{\mathfrak{m}})$. What is an example of a connected commutative ring $A$ which ...
0
votes
1answer
46 views

A nonregular local ring [duplicate]

Consider the ring of the formal power series $k[[T_1,\ldots,T_n]]$ ($k$ algebraically closed) where $\mathfrak m$ is the maximal ideal. If $f\in\mathfrak m^2$, why $$\frac{k[[T_1,\ldots,T_n]]}{(f)}$$ ...
0
votes
1answer
71 views

Some residue field

Consider a prime ideal $\mathfrak{p}\in\mathrm{Spec} \ \mathbf{Z}[x]$; the residue field at $\mathfrak{p}$ is the fraction field of $\mathbf{Z}[x]/\mathfrak{p}$. Can we classify the residue fields? I ...
0
votes
0answers
101 views

Multiplicity of a module in a particular case

We define multiplicity of a module M of dimension $d>0$ as $$mult(M) := lc (P_M) (d-1)!$$ where $P_M$ denotes the Hilbert polynomial of M. Equivalently, we have $mult(M) = Q_M(1)$, where $HP_M (z) ...
1
vote
1answer
42 views

Another description of injective hull

Let $I$ be an injective module containing a module $M$, let $M_1$ be a submodule of $I$ maximal with respect to the property that $M_1∩M=0$, and let $M_2$ be a submodule of $I$ containing $M$ maximal ...
1
vote
0answers
47 views

Deduction of usual Cayley-Hamilton Theorem from “Determinant Trick”

Here is a statement of a standard theorem in commutative algebra (see page 60 of this book): Theorem. ("Determinant Trick") Suppose that $R$ is a commutative ring with $1$. Let $M$ be a finitely ...
2
votes
0answers
66 views

If $R$ is a domain and $M$ a finitely generated $R$-module, is it true that $\bigcap_{f\in M^{*}}\ker{f}=\operatorname{Tor}M$?

Let $R$ be a domain and $M$ a finitely generated $R$-module. Let $M^{*}=\hom_{R}(M,R)$. Let Tor$M$ be the torsion submodule of $M$. It it true that $$\displaystyle\bigcap_{f\in ...
3
votes
2answers
89 views

On Hilbert's Nullstellensatz Theorem

I was reading Ravi Vakil's notes on his website and he states the Hilbert Nullstellensatz (3.2.5.): If $k$ is any field, every maximal ideal of $k[x_1, ..., x_n]$ has residue field a finite extension ...
2
votes
1answer
42 views

Maximal homogeneous ideals of a graded $k$-algebra.

Let $k$ be an algebraically closed field, and let $A$ be a finitely generated commutative $k$-algebra. Given any maximal ideal $\mathfrak{m}\subset A$, we can form the quotient to obtain a map $A\to ...
0
votes
2answers
59 views

Cohen-Macaulay and regularity

I know this is a simple question but to make sure....: $A$ is a commutative ring which is Cohen-Macaulay and for every maximal ideal $\mathfrak{m}$ in $A$ if $\dim A_{\mathfrak{m}}=\dim A$ then ...
3
votes
2answers
38 views

Kahler differentials and quotient rings.

I am dealing with some nice rings that are always isomorphic to some fairly nice quotient ring of a polynomial ring. A typical example is: $$ \mathbb{C}[X,XY,XY^2] \cong ...
0
votes
1answer
92 views

Exercise about intersection of prime ideals

I need to write the ideal $I= (xz-y^2, yt- z^2) \subset R = \mathbb{K}[x,y,z,t]$ as intersection of prime ideals. Any idea? For the moment, I've noticed that I is radical, then it suffices to find ...
0
votes
1answer
68 views

defining the group law on elliptic curves in general

Let $k$ be an arbitrary field and $C \subset \mathbb{P}^2(k)$ an elliptic curve. In order to define the group law on $C$ we need to establish some geometric facts first, e.g. Any line intersects $C$ ...
1
vote
0answers
50 views

Krull dimension of localization

If $R$ is a commutative ring and $m$ a maximal ideal therein, then what are the conditions for the Krull dimension of $R$ equaling to the Krull dimension of $R_m$?
1
vote
3answers
100 views
+50

Some practical questions on cohomology and the ring $\mathbf{Z}[x]/(x^2)$

So I know that the cohomology ring of $S^n$ is $\mathbf{Z}[x]/(x^2)$ with "$x$ in degree $n$"; if we ignore the grading then this ring fails to distinguish the spheres. What is actually meant by "in ...
0
votes
1answer
68 views

Canonical ring map

Let $\chi:\mathbf{Z}\rightarrow A$ be the canonical map to a ring $A$, and let $p$ be a prime ideal of $A$. Then I claim that $\chi^{-1}(p)=(\mathrm{char} \ k(p))$ where $k(p)$ is the residue field at ...
1
vote
1answer
47 views

$\dim(R/x) = \dim(R)-1$ for Noetherian integral domains?

Let $R$ be a Noetherian integral domain of finite Krull-dimension and $0 \neq x \in R$ a non-unit. Do we have $\dim(R/x) = \dim(R) -1$ in general? If this is wrong, does it change something if we ...
1
vote
0answers
19 views

maximal chains of graded prime ideals

In Theorem 1.5.8 [Bruns,Herzog - Cohen-Macaulay-Rings] it is proved that for a noetherian graded ring $R$, a finitely generated graded $R$-module $M$ and any chain $\mathfrak{p}_0 \subsetneq ...
0
votes
1answer
35 views

Example of Tor-Rigid Module

Let $R$ be a commutative ring (with 1) and $M$ a finitely generated $R$-module. We say that $M$ is rigid if for every finitely generated $R$-module $N$ whenever Tor$_i^R(M,N)=0$ then Tor$_j^R(M,N)=0$ ...
3
votes
1answer
49 views

How to characterize all finite commutative local rings with the maximal ideal of fixed order (if the order is small)?

Let $R$ be a finite commutative local ring with the maximal ideal $M$ of order $m$. How to characterize all such finite commutative local rings? For examples, if $m=2$, then $R\cong\mathbb{Z}_4$ ...
2
votes
2answers
76 views

Atiyah & Macdonald Prop. 2.9

The following simple claim is used without proof in Prop. 2.9 of Atiyah & MacDonald (p.23). Although I believe I can prove it with a fairly involved argument, the claim is treated by the authors ...
1
vote
0answers
22 views

Krull dimension and Hilbert polynomial of graded rings

Let $P \subseteq \mathbb{R}^n$ be a polytope with integral vertices, and $Q := \mathbb{R}_+ (P \times \{1\}) \cap \mathbb{Z}^{n+1}$ be the monoid of all lattice points of the cone generated by $P$ in ...
4
votes
1answer
65 views

$(x,y)$-primary ideals

I want to find all ideals $I$ in $\mathbf{C}[x,y]$ with $\sqrt{I}=(x,y)$ and $\dim_{\mathbf{C}}\mathbf{C}[x,y]/I=2$. I have no clue how to about it, I mean I can write down some examples, ...
1
vote
1answer
30 views

Generators for a finitely generated graded ring

Given a Noetherian graded ring (commutative and with 1) $A=\bigoplus_{n=0}^\infty A_n$, that's generated as an $A_0$-algebra by $x_1,\ldots, x_s\in A$. I am having difficulties seeing why there is no ...
2
votes
2answers
76 views

When a finite local ring $R$ has $-1$ as a square in $R^\times$?

Let $R$ be a finite local ring with maximal ideal $M$ such that $|R|/|M|\equiv 1\pmod{4}$. Then $-1$ is a square in $R^\times$ (that is, there exists $u\in R^\times$ such that $u^2=-1$) if and only ...
0
votes
1answer
45 views

An injective-injective module problem

I want to prove this problem: For an $R$-module $M$ and an ideal $J⊆R$, let $A=\{m∈M∶mJ=0\}$. If $M$ is an injective $R$-module, show that $A$ is an injective $R/J$-module. We can view $A$ as an ...
1
vote
0answers
54 views

Tensor product of free modules over free algebra

Suppose $M$ and $N$ are modules over a (commutative, unital) ring $S$. Let $R$ be a subring of $S$ such that $S,M$ and $N$ are all free, finitely generated modules over $R$. Question: Under what ...
0
votes
1answer
35 views

Does the relation $\pi(S_{i})=S^{-1}R-P_{i}\cdot S^{-1}R$ hold for prime ideals $P_i$ in a commutative ring $R$?

Let $R$ be a commutative ring. Let $P_{i}$, $1\leq i\leq n$ be prime ideals none of which are contained in each other. Let $S=R-(\cup_{i=1}^{n} P_{i})$. Then $S$ is a multiplicatively closed set and ...
5
votes
1answer
133 views

The lattice points of a f.g. rational cone form a f.g. monoid.

In their book "Polytopes, Rings, and K-Theory" Bruns and Gubeladze sketch an alternative approach to Gordan's Lemma, which is stated in the headline (f.g. = finitely generated). I don't understand the ...
2
votes
1answer
49 views

Proving that a field $K$ can be generated by algebraically independent elements and an separable element

Let $k$ be a perfect field (either $k$ has characteristic $0$, or characteristic $p > 0$ and every element has a $p$th root), and let $K$ be a finitely generated extension field. I have a question ...
1
vote
2answers
135 views

Is $\mathbb{C}[x,y] / (y^2-x^3)$ a PID?

First, I'd like to show $\mathbb{C}[x,y] / (y^2-x^3)$ is an integral domain. Then I need to find out whether or not it is a PID. For the first part, I want to show $y^2-x^3 \: | \: fg \implies ...
6
votes
5answers
302 views

A finite commutative ring with 1 whose elements satisfy a particular equation

I would be very grateful if you give me a hint on it: Suppose $R$ is a finite commutative ring with identity such that $ x^3 = x $ for all elements $x$ of $R$. Then $R$ is a finite direct product ...
3
votes
1answer
71 views

Universal property of polynomial ring in $\mathbf{CRING}$

I know that the polynomial ring $A[x]$ is the free $A$-algebra on $\{x\}$; this is its universal property in the category of $A$-algebras. Is there also a universal property for $A[x]$ considered as a ...
3
votes
2answers
75 views

How to show this ring is not a UFD

I am trying to show that $R=\Bbb Z[x,y,z,w]/(xw-zy)$ is not a UFD. Let $I=(xw-zy)$. Let $X=x+I$, $Y=y+I$, $Z=z+I$, and $W=w+I$. My guess is that $X$ is irreducible and therefore $(X)$ is a ...
2
votes
1answer
86 views

Atiyah-Macdonald Exercise 2.15

I have worked out a solution to exercise 2.15 of Atiyah-Macdonald, which is needed in the solution of 2.3 (see Atiyah-Macdonald 2.3). However, the solution seems overly complicated, and I am not ...