Questions about commutative rings, their ideals, and their modules.
0
votes
2answers
74 views
Valuation but not Noetherian Rings
For valuation rings I know examples which are Noetherian.
I know there are good standard non Noetherian Valuation Rings. Can anybody please give some examples of rings of this kind?
I am very ...
1
vote
1answer
56 views
Looking for a “prime-ish” family of subsets
Is there a nontrivial (what I mean is below) example of a compact Hausdorff space $X$ and a family $\mathscr{F}$ of subsets of $X$ with the following pair of properties?
$\mathscr{F}$ is ...
1
vote
1answer
65 views
Going down theorem
Here $A$ is a commutative ring with unity.
How to show that going down theorem holds for $A$ contained in $A[x]$, the polynomial ring.
Lying over is ok. I cannot do the other part.
3
votes
2answers
94 views
Localization over commutative Noetherian rings
Let $S$ be a multiplicatively closed subset of a commutative noetherian ring $A$. Let $M$ and $N$ be finitely generated $A$-modules. If $M_S$ is isomorphic to $N_S$, show that $M_t$ is isomorphic to ...
12
votes
5answers
274 views
In a principal ideal ring, is every nonzero prime ideal maximal? [duplicate]
Inspired by this question, I was wondering whether from just the hypothesis that $A[X]$ is a nontrivial (commutative) principal ideal ring (so without supposing it is a domain) one can deduce that $A$ ...
4
votes
1answer
39 views
Do Groebner bases give the smallest generating set for Ideals?
Given a Reduced Groebner Basis $(f_1,\ldots,f_n)$ for an ideal $I$, can there be another basis $(g_1,\ldots,g_m)$ for $I$ where $m<n$?
I've been reading through Cox, but can't seem to find an ...
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 ...
1
vote
2answers
32 views
$k[x^2,x^3]/p$ ($p$:nonzero prime) is integral over $k$?
Let $p$ be a nonzero prime ideal of $A=k[x^2,x^3]$. I want to show $p$ is maximal.
My trial is that $A/p$ contains $k$ and since $k$ is a field, if I can show that $A/p$ is integral over $k$ then it ...
21
votes
0answers
332 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 ...
2
votes
0answers
37 views
counting zeros of complex functions
I'm trying to solve the following question : If $f(z,w)=z^2-w^m$ and $g(z,w)=z^2-w^n$, then $O_2/(f)\cong O_2/(g)$ iff $n=m$. $O_2$ is the ring of all holomorphic functions about zero. One way is ...
2
votes
1answer
48 views
Poles of formal power series (Hilbert-Poincaré series)
How are poles and orders of poles of formal power series defined?
The particular case, I am interested in, is the following definition from [Atiyah-Macdonald, Introduction to commutative algebra, ...
-2
votes
1answer
66 views
How I can demonstrate that f is surjective?
If $f:A\rightarrow B$ is a ring homomorphism that for every prime ideal $p\subseteq A$ the homomorphism $f_{p}:A_{p}\rightarrow B_{p}$ is surjective, then $f$ is surjective.
5
votes
2answers
104 views
Ideals in a Dedekind domain localized at a prime ideal
Let $R$ be a Dedeking domain, let $\mathfrak{i}$ be a non-zero ideal of $R$. By factorization theorem we can write
$$\mathfrak{i}=\mathfrak{p}_1^{a_1}\cdots\mathfrak{p}_n^{a_n}$$
for distinct non-zero ...
1
vote
0answers
52 views
Question about the proof of the going-up theorem of Cohen-Seidenberg.
Let $S$ be a subring of $R$ such that $R$ is integral over $S$. Let $P$ be a prime ideal of $S$ and $M=S-P$. Let $S_M$ be the quotient ring of $S$ and $R_M$ the quotient ring of $R$. Let $i: S \to ...
4
votes
1answer
80 views
Associated Primes of Tensor Product
Let $R$ be a Noetherian ring, and let $M$ and $N$ be finitely generated $R$ module. Do we know any formulas for $\operatorname{Ass}(M\otimes_R N)$ in terms of $\operatorname{Ass}(M)$, ...
3
votes
1answer
41 views
Question about algebraically independence.
Let $R=k[Y_1, \ldots, Y_m]/P$, where $k$ is a field and $P$ is a prime ideal of $R$. Suppose that $Y_1, \ldots, Y_m$ are algebraically independent over $k$. Let $y_1=Y_1+P, \ldots, y_m=Y_m+P$. Can we ...
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 ...
4
votes
1answer
36 views
Isomorphism or non-isomorphism of two specific local rings
Let $K$ be a field and set $A=K[X,Y]/(XY)$ and $B=K[X,Y]/(Y^2-X^3-X^2)$.
Are the two local rings $A_{(X,Y)}$ and $B_{(X,Y)}$ isomorphic?
I think that they are non-isomorphic but I can't prove ...
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 ...
2
votes
1answer
62 views
Show that $K[x,xy,xy^2,\dots]$ is not Noetherian [duplicate]
Here is the problem I am stuck on: Fix a field $K$ and consider the subring $A \leq K[x,y]$ generated by $K \cup \{x,xy,\dots,\}$. Show that $A$ is not Noetherian.
I figure that taking ideals $I_n = ...
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 ...
4
votes
2answers
66 views
Identifying the ideal generated by the variety $V(y^2-x^3)$
I am having trouble showing the following result:
Suppose that $k$ is an infinite field and consider the affine variety $V(y^2-x^3)$. If $I(V)$ denotes the ideal of all polynomials vanishing on ...
3
votes
2answers
46 views
Irreducibility is preserved under base extension
I want to prove that if $A$ is a finitely generated $k$-algebra ($k$ is a field) with prime nilradical then for any field extension $k\rightarrow K$, the $K$-algebra $A\otimes_kK$ has also prime ...
6
votes
1answer
41 views
Verifying Hilberts Nullstellensatz on a particular example
Let $k$ be an algebraically closed field of characteristic $2$ and consider the following equations:
$$xy + z^2 = 0$$
$$uv + w^2 = 0$$
$$uy + vx = 0$$
It's not hard to parameterize solutions to these ...
3
votes
1answer
84 views
Koszul complex of locally free sheaves
Let $X$ be a complex variety; one can also assume it is smooth if this helps. $\mathcal{E}$ is a locally free sheaf of rank $r$ on $X$, and $s \in H^0(X, \mathcal{E})$. Then one has a Koszul complex ...
3
votes
1answer
96 views
Question about isomorphism of modules.
I have been reading the book of DeMeyer and Ingraham "Separable Algebras of Commutative Rings," where in page 129 they prove the following.
Let
$\bullet$ Let $S$ be a commutative ring and $G$ be a ...
2
votes
1answer
66 views
Associated Prime Ideals in a Noetherian Ring
Let $I$ and $J$ be ideals of a Noetherian ring $A$. Then if $JA_P\subset IA_P$ for every $P\in \operatorname{Ass}_A(A/I)$, then $J\subset I$.
I'm reading Matsumura's Commutative Ring Theory book ...
6
votes
1answer
59 views
Is $\operatorname{Tor}_i(M,N)$ of finite length?
Let $A$ be a regular local ring, and let $M$ and $N$ be two finitely generated $A$-modules such that $M\otimes N$ is of finite length, and let $i$ be the largest integer such that ...
2
votes
1answer
47 views
A necessary and sufficient condition for a full lattice over an integral domain
I'm learning about lattices over integral domains and I would be grateful if someone could clarify the following for me.
Let $R$ be an integral domain with quotient field $K$ where $K\neq R$. Suppose ...
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} ...
4
votes
1answer
67 views
Characterization of faithfully flat homomorphisms
Let $A \to B$ be a homomorphism of commutative rings. Why are the following conditions equivalent?
$A \to B$ is faithfully flat.
$A \to B$ is injective, flat and $B/A$ is a flat $A$-module.
This ...
13
votes
2answers
146 views
Is a linear combination of minors irreducible?
Let $X=(X_{ij})_{1\le i,j\le n}$ be a matrix of indeterminates over $\mathbb C$. For choices $I,J\subseteq\{1,\ldots,n\}$ with $|I|=|J|=k$ denote by $X_{I\times J}$ the matrix $(X_{ij})_{i\in I,j\in ...
14
votes
0answers
105 views
Hilbert's original proof of basis theorem
Does anyone know Hilbert's original proof of his basis theorem--the non-constructive version that caused all the controversy? I know this was circa 1890, and he would have proved it for ...
2
votes
1answer
61 views
Homogeneous forms of degree $d$ in quotient ring
We have a nice description for the space of all homogeneous elements of degree $d$ in $R = k[x_1,\ldots,x_{n+1}]$, namely it is isomorphic to $$(x_1,\ldots,x_{n+1})^d/(x_1,\ldots,x_{n+1})^{d+1}.$$
...
2
votes
1answer
59 views
Spectral Sequence involving “Triple Tor”
Can someone help me with the first 4 lines of Page 111 of Local Algebra by Serre?
I would like to know which spectral sequence is being used.
Initially I thought it is the Grothendieck ...
-4
votes
1answer
96 views
A commutative ring whose all localizations are fields
If $A$ is a ring such that $A_{p}$ is a field for every prime ideal $p\subseteq A$, is $A$ a field?
0
votes
0answers
103 views
Exercise about “dimension of rings”
Let $K$ be a field, and $\mathfrak a\subseteq K[X_{1},\dots,X_{n}]$ the ideal generated by the following polynomials of degree one
$$\mathfrak a= \begin{pmatrix}
F_{1}=\sum_{i=1}^{n}a_{1i}X_{i} \\
...
1
vote
1answer
79 views
Exercise about prime ideals in a polynomial ring
Are considered prime ideals $q_{1}\subsetneqq q_{2}\subsetneqq q_{3} \subseteq A[X]$. Could you show that $q_{1}\cap A\neq q_{3}\cap A$ ?
10
votes
1answer
144 views
Geometric meaning of completion and localization
Let $R$ be a commutative ring with unit, $I$ an ideal of $R$ and consider the following three constructions.
The localization $R_I$ of $R$ at $I$ (i.e. the localization of $R$ at the multiplicative ...
5
votes
2answers
91 views
Counterexample for going up theorem
I am searching for an example which shows that integral extensions are necessary for going up theorem.
Basically I want rings $A\subset B$ (not integral extension) such that lying over holds, but ...
0
votes
1answer
29 views
Extension of the radical of an ideal
Let $I$ be an ideal of a ring $R$ and let $f: R\rightarrow S$ be a homomorphism. Let $I^e = \{\sum_{j=1}^n s_j f(r_j) : s_j \in S, r_j \in I\}$ be the extension of $I$. Is it true that the radical ...
1
vote
0answers
29 views
When is the completion of an A-algebra at a height-1 prime just A[[X]]?
Let $A$ be a ring (commutative with unity), and let $B$ be a regular finite-type $A$-algebra of relative dimension 1 over $A$. (ie, Spec $B$ is a regular curve over $A$).
Let $\mathfrak{p}$ be a ...
3
votes
0answers
50 views
primary decomposition of ideals
How to find the primary decomposition of ideal $I = (X^2, XY, XZ, YZ)$ in the ring $k[X,Y,Z]$?
Also how to show that $(X,Y)^{308}$ is primary ideal in $k[X,Y,Z]$? Is there a general rule for finding ...
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 ...
0
votes
1answer
50 views
Localization and initial objects
Let $A$ be a ring and let $S$ be a multiplicative subset of $A$.
Why is the map from $A$ to $S^{-1}A$ initial among all $A$-algebras $B$?
Why does localization not have to commute with respect to ...
1
vote
2answers
89 views
Determine the total ring of fractions
Determine the total ring of fractions of $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}_{12}$.
-1
votes
2answers
116 views
Any subring of $K[X]$ that contains $K$ is noetherian; not all of such subrings are UFDs
Let $K$ be a field. Show that any subring of $K[X]$ that contains $K$ is noetherian. Give an example that demonstrates not all of these subrings are UFDs.
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
53 views
Comparing two expressions of the completion of a ring
Let $A$ be a commutative Noetherian ring and $I=(a_1,\cdots,a_n)$ an ideal. Then the $I$-adic completion of $A$ is isomorphic to $A [[ x_1,\cdots,x_n ]]/(x_1-a_1,\cdots,x_n-a_n)$. Now let $e$ be an ...
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 ...
