2
votes
1answer
28 views

Sets of prime ideal contain a minimal element

I want to prove that every nonempty set of prime ideal contain a minimal element, my attempt is to prove it by using zorns lemma and i would like to know if my proof is valid. Let $\Sigma$ be a ...
1
vote
1answer
26 views

Colon ideal of fractional ideals is itself a fractional ideal

I received this question on homework in my homological algebra class and I need some guidance. Let $R$ be a commutative integral domain and $K$ be its field of fractions. A fractional ideal $I$ of ...
5
votes
1answer
45 views

How to show that $\mathbb{C}[x_1,x_2,x_3, x_4]/(x_1x_2 - x_4x_3, x_1x_3 - x_2x_4, x_4x_1 - x_3x_2)$ is integral domain

I am looking for a way to show that the ring $\mathbb{C}[x_1,x_2,x_3, x_4]/I$ where $I = (x_1x_2 - x_4x_3, x_1x_3 - x_2x_4, x_4x_1 - x_3x_2)$ is an integral domain. In other words I want to show $I$ ...
1
vote
1answer
47 views

A property of minimal prime ideal

Let $R$ commutative ring with unity, $S\subseteq R$ subring, $p$ minimal prime ideal of $S$. Show there exists a minimal prime ideal $q$ in $R$ with the property that the contraction $q^c=q\cap S=p$. ...
2
votes
1answer
45 views

Preimage of maximal ideal is maximal [duplicate]

I've just started a commutative algebra course and I'm stuck on the very first homework problem: Let $A \not= \{0\}$ be a commutative ring. Let $\Phi : A \longrightarrow B$ be a ring homomorphism ...
8
votes
1answer
150 views
+200

Recovering free modules from their projective limit

Let $\dotsc A_2 \to A_1 \to A_0$ be a sequence of surjective homomorphisms of commutative rings. Consider the projective limit $\varprojlim_i A_i$. If $S$ is an (infinite) set, then $\varprojlim_i ...
3
votes
1answer
73 views

Atiyah-MacDonald, Problem 6 of Chapter 1

I was trying to solve the following problem from "Introduction to Commutative Algebra" by Atiyah and MacDonald. (It is Problem 6 of Chapter 1.) While trying to solve the problem, I am facing trouble ...
-2
votes
0answers
41 views

$R_P$ is a valuation noetherian ring

I need a hint to prove this question: Let $R$ be an integral domain and $P\neq 0$ a principal ideal of $R$ such that $\cap_{i=1}^{\infty}P^n=0$, show $R_P$ is a valuation Noetherian ring. I ...
2
votes
4answers
165 views

A finitely dimensional algebra over a field has only finitely many prime ideals all of them are maximal

Let $K$ be a field and let $R$ be a $K$-algebra with unity which is finite dimensional as a $K$-vector space. Prove that $R$ has only finitely many prime ideals all of which are maximal. (Hint: ...
3
votes
3answers
128 views

What are some examples of coolrings that cannot be expressed in the form $R[X]$?

Let $K$ denote a field. Then the polynomial ring $K[x]$ has the property that the sum of two units is either a unit, or zero. I'll bet there's heaps of other examples, though. So let a coolring be a ...
1
vote
1answer
78 views

Ideal generated by a regular sequence

I need to prove that the ideal $$ I = (xz -y^2, x^2t^2 -yz^3, x^2yt^2 -xz^4) \subset R = \mathbb{K}[x,y,z,t]$$ is generated by a $R$-regular sequence. How can I do it? I don't know if this can ...
0
votes
2answers
51 views

Calculating the kernel of a homomorphism

Let $R := k[x, y]$ be a polynomial ring over field $k$. Consider the homomorphism $\lambda : k[x, y, z] \to R \times R$, defined by $\lambda(x) := (x, x)$, $\lambda(y) := (y, y)$ and $\lambda(z) := ...
2
votes
1answer
54 views

Valuation rings of $k(X)$

My question is how to determine all valuation rings of the field $k(X)$ containg the field $k$. I want to show that if $V$ is a valuation ring of the field $k(X)$ and $\neq k(X)$ then ...
4
votes
0answers
72 views

Functorial approach to Ideals and Quotients, Multiplicative Sets and Localizations

I have been playing with substructures of commutative rings today and noticed that there is a strong analogy between the formation of quotients and kernels with the formation of localizations with ...
0
votes
0answers
42 views

If field has a prime field isomorphic to $\mathbb{Q}$, sufficient condition for every subring being integrally closed domain

Suppose that a field $k$ has the prime field isomorphic to the field of rational numbers $\mathbb{Q}$. Then what would be sufficient condition in order for every subring of $k$ be integrally closed ...
4
votes
0answers
63 views

Condition on a field that makes every subring an integrally closed domain

I want to know what condition would need to be additionally imposed on a field to make every subring of the field an integrally closed domain.
1
vote
1answer
48 views

Example of a module such that every proper submodule is finitely generated but the module is not.

Let $R$ be a ring with 1 and $M$ an $R$-module. What is an example such that $M$ is infinitely generated but every proper submodule is finitely generated.
0
votes
1answer
32 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 $k \in \mathbb{N}$, ...
1
vote
1answer
33 views

Ideal quotient and extension

Let $R$ be a commutative ring and $S$ a subring of $R$. If $I$ is an ideal of $S$ define $I^e$ as the ideal in $R$ generated by $I$, i.e. the extension of $I$ in $R$. If $I,J$ are ideals in $S$, we ...
0
votes
1answer
61 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 ...
1
vote
2answers
148 views

Every element in a ring with finitely many ideals is either a unit or a zero divisor.

I came across the above proposition on mathstackexchange If every nonzero element of $R$ is either a unit or a zero divisor then $R$ contains only finitely many ideals. the link asks a different ...
7
votes
3answers
208 views

In $\mathbb{Z}/(n)$, does $(a) = (b)$ imply that $a$ and $b$ are associates?

[Update: Based on the hints provided by @zcn and @whacka, I believe I have found a solution. See my answer below.] Below, $R$ is a commutative ring with $1$. In John J. Watkins' Topics in ...
1
vote
1answer
80 views

The spectrum of a commutative ring with unity and its “topology”

Let $\operatorname{Spec}(R)$ be the set of prime ideals in the commutative ring with unity $R$, and let $\mathfrak a$ be some ideal. Show that we get a topological space if we define the closed sets ...
2
votes
1answer
49 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.
7
votes
2answers
195 views

Showing that $x^3+y^3+z^3=0$ is not rational

Is there a short proof that $F:x^3+y^3+z^3=0$ in $\mathbf{P}^2$ is not rational, apart from using the genus? Perhaps this is an elliptic curve, so every morphism $\mathbf{P}^n\rightarrow F$ is ...
2
votes
1answer
35 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
50 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
76 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
2answers
65 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
42 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 ...
1
vote
0answers
52 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
118 views

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
72 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
2answers
42 views

Finitely generated ideal with special property

Is there a ring with a finitely generated ideal $I$ which has an infinite subset $M\subseteq I$ such that $M$ generates $I$ but no finite subset of $M$ does it? What I found out: If such a rings ...
3
votes
1answer
61 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 $R\cong\mathbb{Z}_4$ ...
1
vote
1answer
54 views

Injectivity of simple modules

If $R$ is a commutative ring with $1$ having a maximal ideal $m$ such that the local ring $R_m$ is a field, how could one check that $R/m$ is an injective $R$-module? If we want to use Baer Lemma, we ...
2
votes
2answers
82 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
64 views

Rank of a module when the base ring is not a domain

Suppose $R$ is a commutative Noetherian local ring with $1$, which is not a domain. Let $M$ be a (non-free) finite $R$-module. What is meant by rank of $M$ in this case?
0
votes
1answer
36 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 ...
1
vote
1answer
86 views

How do we know that $f(x)\in Y$?

At page 19 in this book $f:X\to Y$ is defined to be $$f(a):=(\tilde\varphi(T_1')(a),\dots,\tilde\varphi(T_n')(a)).$$ To explain the notation above, $X\subseteq \mathbb{A}^m(k)$, $Y\subseteq ...
1
vote
2answers
145 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
306 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
2answers
100 views

How to show that $\Bbb Z[x,y,z,w]/(xw-zy)$ is not a UFD [duplicate]

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 ...
-1
votes
1answer
50 views

Graded ring, and its homogeneous ideals : $ I = \displaystyle \bigoplus_ {n \in \mathbb {Z}} (I \bigcap B_n) $

Let $ B = \displaystyle \bigoplus_{n \in \mathbb {Z}} B_n $ be a graded ring. Let $ I $ be an ideal of $ B $. Why is $ I = \displaystyle \bigoplus_ {n \in \mathbb {Z}} (I \bigcap B_n) $ equivalent to ...
5
votes
3answers
88 views

Finitely generated ideal in Boolean ring; how do we motivate the generator?

This problem is Exercise 11.3 in Atiyah/Macdonald Commutative Algebra. They ask to prove every finitely generated ideal in a Boolean ring is in fact a principal ideal. The question has been answered ...
0
votes
1answer
36 views

Global dimension regular rings of finite type

Have I made an error in my reasoning? If $k$ is a field, $A$ is a commutative regular $k$-algebra of finite type and ${\mathfrak{m}}$ is a maximal ideal in $A$ then since $Ext_{A_{\mathfrak{m}} ...
0
votes
0answers
47 views

Is every local ring the localization of some other ring?

One way of constructing a local ring is to start with any commutative ring, and localize all the elements outside of some maximal ideal (i.e., adjoining inverses to all those elements). But I'm ...
5
votes
1answer
67 views

Irreducibility of some multivariate polynomials

Consider the polynomials $xw-yz\in A[x,y,z,w]$ and $x^n+y^n+z^n\in A[x,y,z]$, where $A$ is a commutative ring. I am curious to know what conditions on $A$ (factorial ring, algebraically closed field, ...
0
votes
2answers
42 views

$R^{(I)} \cong K \oplus H$ where $R^{(I)}$ is free but $K$ is not free

Let $R$ be a commutative ring with unit. Is there an example of a direct sum of $R$-modules $$R^{(I)} \cong K \oplus H$$ where $R^{(I)}$ is free but $K$ is not free ? Clearly $R$ can't be a PID.
1
vote
0answers
85 views

What is $\operatorname{Ass}\operatorname{Ext}^i(M,N)$?

This is exercise 1.2.27 of Bruns-Herzog: Let $R$ be a Noetherian ring, $M$ a finite $R$-module and $N$ an arbitrary $R$-module. Deduce that $\operatorname{Ass}(\operatorname{Hom}_R(M,N)) = ...