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4
votes
2answers
772 views

intersection of finitely many maximal ideals

For what commutative rings with infinitely many maximal ideals we can say that the intersection of any combination of finitely many maximal ideals is not zero? Obviously it holds for Dedekind domains ...
1
vote
1answer
165 views

Does the going-up theorem hold between flat algebras?

Let $R$ be a commutative Noetherian ring with unit and $S$ a flat $R$-algebra. Does the going-up theorem hold between $R$ and $S$?
3
votes
2answers
443 views

An identity in an arbitrary commutative ring

This fact might be either trivial, wrong, or well known. Let $R$ be a commutative ring. Let $u_1,\dots,u_{s-1},u_s\in R$ and $m,M\in R$. Let us assume that $m,M$ satisfy $$(m-u_1) \dots ...
4
votes
2answers
657 views

The Weyl algebra modules which are also rings

Consider the space $C^\infty(\mathbb{R})$. We have on it a natural action of the Weyl algebra generated by $x$ and $d/dx$, where $x$ is the coordinate on $\mathbb{R}$, and there is plenty of Weyl ...
0
votes
0answers
116 views

Cubic field and the corresponding cubic binary form

I am currently reading about binary cubic forms and cubic number fields (mainly about using binary cubic forms with integer coefficients to parametrize orders in the cubic field) and I thought it ...
-4
votes
1answer
107 views

R is a commutative ring and every ideal is either R or 0. Show that R is a field [closed]

Assume that R is a commutative ring and every ideal is either R or 0. Show that R is a field. How can I show this without first being given that $ 1\ne 0 \in R$
0
votes
2answers
263 views

Tensor powers of an algebra all isomorphic

Let $k$ be a commutative ring with total quotient ring $K$, and let $A$ be a commutative $k$-algebra such that the multiplication map $A \otimes_k A \longrightarrow A$ is an isomorphism. EDIT: Assume ...
3
votes
2answers
462 views

Condition for a local ring whose maximal ideal is principal to be Noetherian

The statement "a local ring whose maximal ideal is principal is Noetherian" is (I think) false. The ring of germs about $0$ of $C^\infty$ functions on the real line seems to be a counterexample since ...
1
vote
0answers
92 views

Graded Betti Numbers of a Graded Ideal with Linear Quotients

Exercise 8.8 in Monomial Ideals by Herzog and Hibi: Let $I\subset S=K[x_{1},...,x_{n}]$ be a graded ideal which has linear quotients with respect to a homogeneous system of generators ...
1
vote
1answer
153 views

Do group identities of quotient with radical lift?

Let $R$ be a commutative Artinian ring and $J(R)$ its radical. Assume that the quotient $R/J(R)$ is a GI-ring. (definitions that i use: I call a ring $S$ a GI-ring if its unit group, ...
0
votes
1answer
366 views

Recursive Non-standard Models of Modular Arithmetic? [closed]

Any algebraically closed field (ACF) is a model of Modular arithmetic (MA). (MA) has the same axioms as first order Peano arithmetic (PA) except $\forall x(Sx \neq 0)$ is replaced with $\exists ...
2
votes
1answer
198 views

Checking flatness using radical ideals

Let $R$ be a commutative ring and $M$ a not necessary finitely presented $R$-module. I am looking for a prove or a counterexample to the following statement: $M$ is flat as an $R$-module if and only ...
1
vote
0answers
117 views

Universally catenary and all its formal fibers over minimal members are Cohen-Macaulay but it has a nonCohen-Macaulay formal fiber

Please help me to find a Noetherian local ring $R$ such that: $R$ is universally catenary and all its formal fibers over minimal members of $Spec(R)$ are Cohen-Macaulay but $R$ has a nonCohen-Macaulay ...
6
votes
0answers
169 views

Square of primary ideals

Is there any example of a $P$-primary ideal $I$ in a noetherian domain $R$ such that $I^2=PI \not=P^2$?
4
votes
0answers
250 views

Ext groups of affine scheme

Let $A$ be a commutative ring. $\textbf{Spec}(A)$ is the the spectrum of $A$. $M$ and $N$ are $A$-modules. $\tilde{M}$ and $\tilde{N}$ are sheaves associated on $\textbf{Spec}(A)$ respectively. I ...
0
votes
1answer
179 views

Depth of polynomial ring $S=\Bbb{R}[x_1,x_2,x_3,…,x_n,…]$

Consider the polynomial ring of countable variables with coefficients in the real numbers, i.e, $S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$. My Question is about the depth of this ring. Question: Could we ...
5
votes
3answers
197 views

Subset of Spec(A) realized as inverse image of some Spec(B)

Let $A$ be a (commutative) ring and $U$ be an arbitrary subset of $\mathrm{Spec}(A)$. Do there exist a ring $B$ and a ring homomorphism $\varphi\colon A\to B$ such that ...
1
vote
1answer
302 views

chain of prime ideals in polynomial ring $S=\Bbb{R}[x_1,x_2,…,x_n,…]$

Consider the polynomial ring of countable variables with coefficients in the real numbers, i.e, $S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$. Is the following question true? Question: Is there any maximal ...
9
votes
0answers
197 views

Is the generation of rings by their units a question in K-theory?

Susan's question When can number rings be spanned (as $\mathbb{Z}$-modules) by units? smells like an algebraic K-theory question in disguise. I'll reformulate the question first: Given an integral ...
2
votes
1answer
110 views

Hilbert Regularity in relation to degree of generators

Suppose we look at the $\mathbb {C}$ module $\mathbb{C}[x_1,\dots,x_n]=\oplus{S_i}$ where $S_i$ are the polynomials of degree $i$. Then we look at a subring $R$ (also a $\mathbb {C} $ module) ...
10
votes
2answers
405 views

Are these rings of functions isomorphic?

Let $R$ be the ring of all functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ which are continuous outside $(-1,1)$ and let $S$ be the ring of all functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ which are ...
0
votes
1answer
283 views

Number of Minimal left ideals in the full matrix ring over a finite commutative local ring

Inspired with another QUESTION I would like to know the number of minimal left ideals of $M_n(R)$ in terms of $n$ and $R$ where $R$ is a finite local commutative ring with identity ?
6
votes
1answer
533 views

Is an ideal generated by multilinear, irreducible, homogeneous polynomials of different degrees always radical?

I asked this question on math.se and someone even put a bounty on it, yet there was no answer. Hence, I am asking here. Assume $\Bbbk$ to be a field of characteristic zero. Definition. A ...
2
votes
0answers
54 views

lift isomorphic in a sufficiently thick fiber

Let $P$ and $P'$ two polynomials in $k[[\pi]][t]$ for an algebraically closed field $k$ and let $A=k[[\pi]]$. We consider $ X'=Spec (A[t]/(P'))$ and $X=Spec (A[t]/(P)$. Let $d=val(\Delta(P))$ where ...
2
votes
1answer
98 views

on smoothness of morphisms on an artinian base

Let $A$, $B$ two smooth $R$-algebras of finite type for a artinian local ring $R$. Let $I$ an ideal such that $I^{2}=0$ and $\bar{R}=R/I$. We assume that the map $Spec B/IB\rightarrow Spec A/IA$ is ...
2
votes
0answers
97 views

What is known about the krull dimension of an ultrapower ring?

Let $R$ be a ring, $F$ a free ultrafilter on a set $X$ which is not countably complete, and $R_F$ the ultrapower of $R$ with $R \not\cong R_F$. The following two results are from a masters thesis ...
2
votes
1answer
181 views

elementary question on a completion of a ring

Let $k$ a field, and $k[\epsilon]=k[X]/(X^{2})$ , what is the completion of the ring $k[\epsilon][t]$ with respect to the ideal $(t^{2}+\epsilon)$?
2
votes
1answer
172 views

A class of rings related to rings with IBN property [closed]

A ring $R$ (with 1) has IBN property if free $R$-modules have unique rank (e.g., commutative rings). In the same fashion, lets call $R$ a good ring if in every free $R$-module any independent set can ...
5
votes
2answers
438 views

Isomorphic rings of functions

Let $X$ and $Y$ be two topological spaces with $C(X) \cong C(Y)$ (where $C(X)$ is the ring of all continuous real valued functions on $X$). I know that we can not conclude that $X$ and $Y$ are ...
2
votes
1answer
238 views

Symmetric algebra of an ideal and syzygies

Let $(R,\mathfrak{m})$ be a Cohen Macaulay local ring and $I=(a_1,\ldots,a_g,a_{g+1})$ an almost complete intersection ideal of codimension $g.$ Let $R^k\longrightarrow R^{g+1}\longrightarrow ...
7
votes
3answers
725 views

Idempotent polynomials

Let $R$ be a commutative ring with identity and let $f \in R[x]$. There are well known characterizations for $f$ to be a nilpotent element of $R[x]$ or to have a multiplicative inverse in $R[x]$. Is ...
4
votes
1answer
113 views

Algorithm to detect if an element of a (commutative) ring is in a subring?

For rings finitely-generated over a field, the theory of Groebner bases gives us quite an efficient algorithm for determining whether an element of the ring is in a given ideal of the ring. Is there ...
4
votes
2answers
649 views

Example of a reduced ring whose completion is not reduced

Let $(R,\mathfrak{m})$ be a local Noetherian ring. Give an example such that R is reduced but $\mathfrak{m}$-adic completion of R is not reduced.
1
vote
1answer
133 views

Finding reducible polynomials with restricted factors

Given $f(x),g(x) \in \mathbb{Z}[x]$, two irreducible polynomials, is there a polynomial $h(x) \in \mathbb{Z}[x]$ coprime to $f(x)$ such that $f(x) + g(x)h(x)$ is reducible over $\mathbb{Z}[x]$ with ...
1
vote
2answers
215 views

Decomposition of a quotient module

Let $R=k[v,x,y,z]/I$, with $I=\langle v^2,z^2,xy,vx+xz,vy+yz,vx+y^2,vy-x^2\rangle$,and let $f:R^2 \rightarrow R^2$ denote the map given by the matrix $$M=\begin{pmatrix} v & y \\ x & z ...
0
votes
0answers
128 views

Ring algebraically closed in its completion.

First I would like to be clear about the definition, which I am having trouble finding. What does: The local ring $A$ is algebraically closed in $B\supset A$. (e.g. for $B:=\hat{A}$, the completion ...
2
votes
1answer
188 views

for which truth-operations f can f-membership in a prime ideal be represented by a polynomial?

Nota bene: all rings are supposed commutative with $1$ and all ring homomorphism should be unital. Let $B$ be the set of truth values {True, False}. For a formula $\phi$ denote by $[\phi]\in B$ it's ...
4
votes
1answer
478 views

Surjectivity of the natural map of injective module to its localization

Hello all, The lemma 3.3 page 214 in Hartshorne Algebraic Geometry book states:" If $I$ is an injective module over a Noetherian ring $A$. Then for any $f\in A$, the natural map of $I$ to its ...
0
votes
0answers
86 views

Example of a ring whose minimals are annihilators of idempotents?

I'm looking for examples† of rings with the property that for each $P={\rm Ann}_R(a)\in{\rm Min}(R)$ then $a\in R$ is idempotent (ie $a^2=a$) † other than domains!
6
votes
1answer
463 views

Are roots of transcendental elements transcendental?

This looks extremely easy, but then again it's late at night... Let $k$ be a commutative ring with unity. An element $a$ of a $k$-algebra $A$ is said to be transcendental over $k$ if and only if ...
0
votes
1answer
173 views

Relation between $H^i_I(-)$ and $H^i_J(-)$ when $I\subset J$

What is the relation between $H^i_I(-)$ and $H^i_J(-)$ (cohomological functors) when $I\subset J$ are ideals of a (local) noetherian ring?
5
votes
2answers
386 views

Homological characterization of smooth maps

Let $A \to B$ be a finitely generated homomorphism between two commutative noetherian rings. As far as I understand, in various generalizations of this situation, such a map is called smooth if $B$ ...
1
vote
1answer
126 views

An example of a ring $R$ with the property that for each $P=Ann_R(r)\in {\rm Min}(R)$ we have $Ann_R(P)=Rr$.

I'm looking for an example of a commutative (preferably local) ring $R$ such that ${\rm dim}R>0$ and $R$ has the property that for each $P=Ann_R(r)\in {\rm Min}(R)$ we have $Ann_R(P)=Rr$. This ...
0
votes
1answer
81 views

Explicit representation of $R[\frac{x}{y}]$ where $x, y\in R$ for non-Euclidean PIDs $R$?

It's a fact proven by Pendleton, Gilmer, and Ohm (as an obvious corollary of their work, anyways) that PIDs are QR-domains, meaning every overring (ring between the domain and the quotient field) is a ...
2
votes
3answers
240 views

When does “second annihilator” of a (principal) ideal equal the ideal itself , ie $Ann_R(Ann_R(r))=Rr$?

Suppose that $R$ is a (local) ring and $r\in R$. When do the equations $Ann_R(Ann_R(r))=Rr$ or $\sqrt{Ann_R(Ann_R(r))}=\sqrt{Rr}$ hold? I already know that it holds for Artinian Gorenstein rings (due ...
4
votes
1answer
181 views

Number of generators of colon and intersection ideals of two finitely generated ideals in a Prufer domain

Hi! I'm wanting to see why the following is true: Given 2 finitely generated ideals $B$ and $C$ in a Prufer domain $D$ with bases of $n$ and $m$ generators respectively, $B\cap C$ has a basis of ...
3
votes
1answer
223 views

Example of an overring of an integral domain which is not a ring of quotients?

Hi! I'm trying to make headway on a question for my undergraduate honors thesis, specifically the question of which rings of integer-valued polynomials if any satisfy the QR-property; that is, the ...
4
votes
2answers
468 views

When does End(M) consist entirely of zero, zero divisors, and units?

Let $R$ be a commutative ring (with $1$) such that every non-zero divisor in $R$ is a unit (see Rings in which every non-unit is a zero divisor for various stabs at what these are called). Let $M$ be ...
0
votes
0answers
61 views

Algorithm for computing basis of zero dimensional ring?

If given a zero dimensional ring over a field, for example, a polynomial ring $A=k[x_1,\ldots,x_n]/(f_1,\ldots,f_n)$ such that $A$ is 0-dimensional, is there an algorithm to compute a monomial basis ...
2
votes
3answers
2k views

Why Does Induction Prove Multiplication is Commutative?

Andrew Boucher's General Arithmetic (GA2) is a weak sub-theory of second order Peano Axioms (PA2). GA has second order induction and a single successor axiom: $$\forall x \forall y \forall ...