Tagged Questions
4
votes
3answers
63 views
example of a flat but not faithfully flat ring extension
I am learning commutative algebra and there is a definition about faithfully flat modules or ring extensions. I can't think of an example of a flat but not faithfully flat ring extension or module. ...
8
votes
2answers
60 views
If $R$ is a commutative ring with unity and $R$ has only one maximal ideal then show that the equation $x^2=x$ has only two solutions
If $R$ is a commutative ring with unity and $R$ has only one maximal ideal then show that the equation $x^2=x$ has only two solutions.
I know that $0$ and $1$ are the solutions, but I can't proceed ...
4
votes
0answers
54 views
Artinian rings are perfect
Is there a simple way to prove that an Artinian ring is perfect? (in the commutative case)
7
votes
2answers
53 views
Why over $\mathbb{Z}/n\mathbb{Z}$ projectivity, injectivity and flatness coincide for cyclic modules?
Assume $R=\mathbb{Z}/n\mathbb{Z}$ ($n\neq0$) and let $M$ be a cyclic $R$-module. Could you tell me how to prove that $M$ is projective if and only if it is injective if and only if it is flat? And ...
3
votes
2answers
63 views
A noetherian ring whose ideals are idempotent is artinian
I have to prove the folowing:
If $R$ is a Noetherian ring, and for every ideal $I$ of $R$ we have $I = I^{2}$, then $R$ is Artinian.
My first thought was to try to prove that the nilradical of ...
5
votes
0answers
87 views
An example of a commutative ring in which every primary ideal is prime
It is clear that every prime ideal in a commutative ring is primary. The converse is false; for example, in the ring $\mathbb{Z}$ the ideal $(p^2)$ is an example of a primary ideal that is not prime ...
5
votes
1answer
50 views
What can be said about $p\in Spec(R)$ when $R_p$ is a field?
What can be said about $p\in Spec(R)$ when $R_p$ is a field? Especially when $R$ is local noetherian
3
votes
0answers
52 views
Integral homomorphism induces a closed map on spectra
I'm trying to prove the following:
Let $f:A\rightarrow B$ is a integral homomorphism (e.g. $B/f(A)$ is a integral extension). Consider $f^{*}: \operatorname{Spec}B \rightarrow ...
2
votes
3answers
60 views
Spectrum of polynomial ring
In M. Reid's Undergraduate Commutative Algebra, the author states that if $k$ is an algebraically closed field then $\operatorname{Spec}{k[x]} = \{0\} \cup k$ (page 21). Is this correct? Instead, ...
2
votes
1answer
41 views
Is localization of a prime ideal still a prime ideal?
Im still new to the topic so this question might seem trivial. But I hope if someone can help explaining to me if a prime ideal $P$ of a domain $A$ is still a prime ideal $P_s$ in the localization ...
10
votes
2answers
80 views
What is a primary decomposition of the ideal $I = \langle xy, x - yz \rangle$?
Given the ring $k[x,y,z]$, where $k$ is a field, and an ideal $I=(xy,x-yz)$, find the primary decomposition of $I$.
I tried to draw the graph of the variety of $I$ and get a decomposition of ...
10
votes
1answer
158 views
Classification of local Artin (commutative) rings which are finite over an algebraically closed field.
A result in deformation theory states that if every morphism $Y=\operatorname{Spec}(A)\rightarrow X$ where $A$ is a local Artin ring finite over $k$ can be extended to every $Y'\supset Y$ where $Y'$ ...
7
votes
1answer
49 views
When does “second annihilator” of a (principal) ideal equal the ideal itself
Suppose that $R$ is a (local) ring and $r\in R$. When do the equations $\operatorname{Ann}_R(\operatorname{Ann}_R(r))=Rr$ or $\sqrt{\operatorname{Ann}_R(\operatorname{Ann}_R(r))}=\sqrt{Rr}$ hold?
I ...
1
vote
0answers
42 views
$0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ exact, $M''$ flat. Why is $M$ flat $\Leftrightarrow M'$ flat?
Let $A$ be a commutative ring with identity, let
\begin{align}
0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0
\end{align}
be an exact sequence of $A$-modules, let $M''$ be flat.
I want ...
1
vote
1answer
62 views
Localization of $K[x,y|x^2-y^3]$ and $K[x,y|xy]$ at $\langle x,y\rangle$ and $\{\text{non-zero-divisors}\}$ (exercise in SICA)
In Greuel & Pfister's A Singular Introduction to Commutative Algebra, p. 38, there is written:
So we have rings
$$\begin{array}{l l}
R_1:= K[x,y|x^2\!-\!y^3], & R_4:= K[x,y|xy],\\
R_2:= ...
1
vote
0answers
44 views
Noetherian localizations and extra-condition implies Noetherian
I'm trying to solve this question but I'm stucked:
If a ring $R$ satisfies the following two conditions:
i) For every maximal ideal $M$ of $R$, if $S = R\setminus M$ then $S^{-1}R$ is ...
3
votes
1answer
43 views
Difficulty Understanding Primary Modules
I have read that any irreducible sub-module $I$ of a Noetherian module $M$ is primary. However if we let $M = \mathbb{Z}_8$ and $I = \mathbb{4Z}_8$ this isn't true, because $I$ is irreducible, and ...
3
votes
1answer
95 views
About Artinian Rings
I'm studing commutative algebra by the text of Atiyah and Macdonald, and a doubt come at me and I can not prove neither find a counterexample, the problem is:
If a ring (commutative with identity) ...
5
votes
2answers
72 views
Algorithmic approach to enumerating ideals in $\Bbb Z[x]/(m, f(x))$
I'm studying for my algebra quals this fall and keep encountering problems like the following:
List all the ideals of $\mathbb{Z}[x]/(16, x^3)$.
or
List all the primes of ...
1
vote
1answer
85 views
Artinian ring and faithful module of finite length
Let $A$ be a ring. How can I prove that:
$A$ is an Artinian ring $\Leftrightarrow \exists$ a faithful $A$-module which is of finite length.
I know that if a ring has a faithful $A$-module which ...
1
vote
0answers
33 views
Equality of two $k$-algebras
Let $f\in k[X_1,\ldots, X_n]$ and $1-fX_{n+1}\in k[X_1,\ldots, X_{n+1}]$. Moreover $X\subseteq k^n$ is a subset and
$$I(X)=\{g\in k[X_1,\ldots, X_n]\,:\, g(x)=0\,\forall x\in X \}$$
is the ideal of ...
3
votes
1answer
43 views
$S^{-1}B$ and $T^{-1}B$ isomorphic for $T=f(S)$
Let $f:A\to B$ be a homomorphism of rings, $S$ be a multiplicatively closed subset of $A$ and $T=f(S)$. Then $S^{-1}B$ and $T^{-1}B$ are isomorphic as $S^{-1}A$-modules.
First we define the ...
0
votes
3answers
71 views
Integral Dependence & Finitely Generated Modules
How to prove $(3)\Rightarrow(1)$ of this theorem:
Let $A\subseteq B$ be commutative rings. The following are equivalent:
$(1)~~x\in B$ is integral over $A$;
$(2)~A[x]$ is a finitely generated ...
4
votes
1answer
86 views
Field of fractions of $\mathbb{Q}[x,y]/\langle x^2+y^2-1\rangle$ [duplicate]
This problem goes as follows:
Prove that $\mathbb{Q}[x,y]/\langle x^2+y^2-1\rangle$ is an integral domain and that its field of fractions is isomorphic to the ring of rational functions ...
4
votes
0answers
52 views
A question on an answer on Math Overflow about Artin approximation
I have a question on an answer of this Math Overflow question.
Let $(A,I)$ be a commutative excellent normal local domain. The completion
$$
\hat A=\underset{\longleftarrow}{\operatorname{lim}} ...
2
votes
1answer
56 views
Noetherian and Artinian modules over subrings
I have a question about whether Noetherian-ness and Artinian-ness of modules are preserved under changes of the base ring. More precisely:
Let $R$ be a commutative ring and $S \subseteq R$ a ...
2
votes
1answer
37 views
The local cohomology modules are Artinian
Let $(R,m,k)$ be Noetherian local ring and $M$ a finitely generated $R$-module. Lemma 3.5.4 of Bruns-Herzog states that
the local cohomology modules $H^i_m(M)$ are Artinian
and that this ...
2
votes
1answer
33 views
The set of zero divisors is the union of radicals of annihilators
I am trying to figure out why the statement
$$\text{the set of zero divisors }=\bigcup_{0\ne x\in R} \sqrt{\text{Ann}(x)}$$
is true. Here $R$ is a commutative ring, $\text{Ann}(x)=\{r\in R\mid rx=0\}$ ...
6
votes
4answers
88 views
Definition of Jacobson radical
This may be a rather silly question, but I wonder why the definition of the Jacobson radical always is
$$\{x\in R\mid 1-xy \text{ is a unit for all } y\in R\}$$
and not
$$\{x\in R\mid 1+xy \text{ is ...
2
votes
1answer
33 views
On regular elements and Maximal Cohen-Macaulay modules
I was reading theorem 3.3.3 in Bruns-Herzog: we have a Cohen-Macaulay local ring $(R,\mathfrak m,k)$, $C$ and $M$ are maximal Cohen-Macaulay modules. (Probably to solve my question some of these ...
3
votes
0answers
55 views
Hilbert’s zeros theorem, an application. (The algebraic variation)
Theorem: (Hilbert) If $k$ is a field, $A$ is a finitely generated $k$-algebra, and $M$ is a maximal ideal in $A$, then the factor $A/M$ is a finite extension of $k$. In particular if $k$ is ...
0
votes
1answer
29 views
If $x_i$ generate an $A$-module $M$, why do $1 \otimes x_i$ generate the extension of scalars $B \otimes_A M$?
In the following, let "ring" be a synonym for "commutative ring with identity".
For rings $A, B$ and an $A$-module $M$, let $M_B = B \otimes_A M$ be the $B$-module obtained from $M$ by extension of ...
4
votes
1answer
61 views
Why are quotient modules $M / \mathfrak{m}M$ over residue fields $A / \mathfrak{m}$ considered for local rings rather than general rings?
In the following, let "ring" be a synonym for "commutative ring with identity". In the book on Commutative Algebra by Atiyah and MacDonald, I read:
Let $A$ be a local ring, $\mathfrak{m}$ its ...
1
vote
2answers
43 views
A complete set of orthogonal idempotents in a commutative ring
I'm reading David Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry. At page 13, Chapter $0$, he says: "... if $e_1,\ldots,e_n$ is a complete set of orthogonal idempotents in a ...
0
votes
0answers
77 views
How to show an ideal is zero-dimensional? [duplicate]
Let $J$ denote the ideal in $\mathbb{Q}[x,y,z]$ generated by $\{y^2-xy-2xz,y^3+z^2+1, x^2yz-yz\}$. Show that $J$ is zero-dimensional.
How do I go about showing this?
5
votes
1answer
49 views
For a group-algebra $k[G]$ ($G$ finite), why is a $k[G]$-module the same as a $k$-representation of $G$?
I'm reading the Atiyah-MacDonald book on Commutative Algebra.
At the beginning of the module chapter on page 17, they make an example which I don't understand. Example 5) is:
$G$ = finite group, ...
8
votes
1answer
80 views
Commutativity characterization?
Let $R$ be a ring (not necessarily unital) and for any $x\in R$ there is an integer $n \geq 2$ s.t. $x=x^2+\cdots+x^n.$
Does it imply that $R$ is commutative?
1
vote
1answer
58 views
Krull Dimension
I'm studying Krull dimension and I'm confused about the definition of $\text{ht}(P)$, which is as I understand is the following: let $$P_0\subset P_1\subset\dots\subset P_n=P$$ be a chain of prime ...
3
votes
1answer
43 views
Correspondence between submodules and quotient modules
What is the (natural) bijection between the set of all sub modules upto isomorphism and set of all isomorphic quotient modules upto isomorphism of a finitely generated torsion module over a PID. Is ...
2
votes
1answer
32 views
About injectivity of induced homomorphisms on quotient rings
Let $A, B$ be commutative rings with identity, let $f: A \rightarrow B$ be a ring homomorphism (with $f(1) = 1$), let $\mathfrak{a}$ be an ideal of $A$, $\mathfrak{b}$ an ideal of $B$ such that ...
21
votes
0answers
333 views
A short proof for $\dim(R[T])=\dim(R)+1$?
If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and nontrivial ...
1
vote
1answer
28 views
Finitely generated integral domain and finitely generated $k$-algebra.
Let $k$ be a field and $R$ a finitely generated domain over $k$. Then there are $y_1, \ldots, y_n$ such that $$R=k[y_1, \ldots, y_n]/P,$$ where $P$ is some prime ideal of $k[y_1, \ldots, y_n]$. My ...
2
votes
0answers
61 views
A commutative ring with alternating and commutativity properties with infinite distinct elements
Is there any nontrivial commutative ring without multiplicative identity that satisfies alternating property ($x \cdot x = 0$ for all $x$ where $\cdot$ is multiplication operator and $x \cdot y \neq ...
1
vote
2answers
63 views
Two principal ideals coincide if and only if their generators are associated
Suppose we have a ring $R$ and $(a),(b)$ are both ideals of $R$. Is it always true that $(a)=(b)$ if and only if there exists a unit $c$ such that $a=bc$ (i.e., $a$ and $b$ are associate)?
I ...
1
vote
1answer
41 views
Integral extension implies that the induced map on prime spectra is closed
Say we have an integral extension $f:R \hookrightarrow S$ of rings. I want to show that the induced map $f^*:Spec(S) \twoheadrightarrow Spec(R)$ is closed. In other words, let $V(I) = \{\mathfrak{P} ...
0
votes
2answers
52 views
Nilradical and Jacobson's radical. [duplicate]
Let A be a commutative ring with 1.
1) Prove that a sum of a nilpotent element and an invertible element is invertible.
2) Prove that if $f=a_0+a_1x+\dots+a_nx^n \in A[x]$
a) $\exists f^{-1}\in ...
1
vote
1answer
38 views
Examples of $I$-adically incomplete rings
I seem to be short on examples for $I$-adic completions of rings.
I know that a ring is $I$-adically complete if the canonical homomorphism into the inverse limit is an isomorphism. My thinking and ...
2
votes
1answer
65 views
If $B$ is finitely generated as a $k$-algebra, and $\phi:A\to B$ is a $k$-algebra map, is $\phi^{-1}(M)$ maximal for any maximal $M\subset B$?
Suppose that $A$ and $B$ are commutative rings containing a field $k$, and $B$ is finitely generated $k$-algebra. Let $\phi: A\rightarrow B$ be a ring homomorphism with $\phi|_k =\mathrm{Id}$. I am ...
-1
votes
1answer
31 views
subring isomorphic to the ring and surjection
Let $f : A \to B$ be a homomorphism of unital commutative artinian rings.
If $f^{-1}(B)$ is isomorphic to $A$, is $f$ surjective ?
6
votes
2answers
92 views
Is $(R_S)_{\mathfrak{p}R_S}$ isomorphic to $R_{\mathfrak{p}}$?
Let $R$ be an integral domain, let $S$ be a multiplicative subset of $R$, not intersecting $\mathfrak{p}$, where $\mathfrak{p}$ is a prime ideal of $R$. Hence $\mathfrak{p}R_S$ (the ideal generated by ...
