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3
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1answer
406 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
1answer
152 views

Uniform Artin-Rees

The Artin-Rees lemma states that if $R$ is a Noetherian ring, $I \subseteq R$ is an ideal and $N \subseteq M$ are finitely generated $R$-modules, then there exists an integer $k$ such that for every ...
8
votes
0answers
176 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 ...
7
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0answers
163 views

Square of primary ideals

Is there any example of a $P$-primary ideal $I$ in a noetherian domain $R$ such that $I^2=PI \not=P^2$?
5
votes
0answers
323 views

Does existence of an isolated solution imply the Jacobian determinant is non-zero?

Let $f_1,\dots,f_n$ be formal power series in $\mathbb{C}[[x_1,\dots,x_n]]$ whose constant terms are all zero (i.e. $f_1,\dots f_n$ are not units in the ring). Suppose further that the radical of the ...
5
votes
0answers
562 views

Maximal ideals of the rings of Baire- One Functions

A real function $f:X\rightarrow \mathbb{R}$ Is called Baire-one function, if there is a sequence $(f_n)_{n=1} ^\infty$ of continuous functions $f_n:X\rightarrow \mathbb{R}$ on $X$ so that for all ...
5
votes
0answers
243 views

Testing isomorphism of finitely generated algebras

Let $A=\mathbf{Q}[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over the rational numbers. Let $B=\mathbf{Q}[f_1,\ldots,f_r]$ and $C=\mathbf{Q}[g_1,\ldots,g_s]$ be two finitely generated ...
5
votes
0answers
856 views

Are local, Noetherian rings with principal maximal ideal PIR?

A question asked by a friend. I believe it's false, but lack a decisive counterexample. This question shows that it is true for valuation rings, but I know too little about them. In the wider ...
4
votes
0answers
226 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 ...
4
votes
0answers
653 views

Commutative ring Notes by M. Artin

In 1966, Professor Michael Artin gave a course for first-year graduate students at MIT on commutative algebra. In that course he covered many classical topics, (the Spectrum of a commutative ring, ...
4
votes
0answers
213 views

When is the ring of invariants of a finite group generated by symplectic reflections a complete intersection ring?

Let V be a finite dimensional symplectic vector space over $\mathbb{C}$. Let $G$ be a finite subgroup of the symplectic group $Sp(V),$ which is generated by symplectic reflections, i.e. by elements ...
3
votes
0answers
84 views

which automorphisms of a subring extend to those of a ring

(Probably a silly question, but..) Consider the ring $R=k[[x_1,\dots,x_n]]/I$, (e.g. char(k)=0), and its subring, $R_1$, generated by some of $x_i$'s. In general, an automorphism of $R_1$ does not ...
3
votes
0answers
142 views

Generic Rank of R^{1/p}

Suppose $R$ is a local Noetherian domain of dimension $d$ in characteristic $p>0$. Suppose $R^{1/p}$ is a finitely generated $R$-module, and suppose $k$ is the residue field of $R$. Is the ...
3
votes
0answers
558 views

Basic commutative algebra question.

Suppose that A is a local ring (commutative with unit), finite over a field k. Let L be the residue field A / m where m is the unique maximal ideal of A. Does the dimension of L (as a k-vector space) ...
2
votes
0answers
70 views

Completion of Bezout Domain a Bezout Domain?

Let $R$ be a Bezout domain, and $I$ any ideal inside of $R$. Is the $I$-adic completion $$ \varprojlim_i R/I^i $$ necessarily a Bezout domain? If not, what conditions (on $R$ or $I$) might ensure that ...
2
votes
0answers
135 views

Domains with prime ideal theorems

Let $D$ be a domain, and for prime ideals $\frak P$ of $D$ the norm is $N({\frak P}):=|D/{\frak P}|$. The prime ideal counting function of $D$ is given by $\pi_D(x)=\#\{{\frak P}\in{\rm ...
2
votes
0answers
100 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 ...
2
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0answers
63 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) ...
2
votes
0answers
52 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
0answers
67 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
0answers
104 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 ...
2
votes
0answers
274 views

Orthogonality (wrt. Ext, Tor) in commutative noetherian rings

Hi, it is a folklore, that: let $p$, $q$ be two primes of a commutative Gorenstein ring $R$. $$ \operatorname{Tor}^k(E(R/p), E(R/q)) \neq 0 \iff p = q\mbox{ and }k = \operatorname{height} p. $$ ...
2
votes
0answers
461 views

Decomposition group vs Galois group of completed extension for height > 1 primes

Assume Let $R$ be a Noetherian normal excellent domain, $F$ its field of fractions. Let $S$ be a finite $R$-algebra, $L$ its field of fractions. $L/F$ a (finite) Galois extension $S$ normal in $L$ ...
1
vote
0answers
126 views

On Prüfer domains

Is there any Prüfer domain $R$ that has a prime ideal $P$ that is not finitely generated but $xP$ is subset of a finitely generated ideal $I$,for some $x$ in $R-P$ and $I$⊂$P$?
1
vote
0answers
45 views

Ideals in a reduced ring

In a finite reduced commutative ring, is every minimal prime ideal is generated by an idempotent? The number of idempotent elements and the number of minimal prime ideals are same.
1
vote
0answers
116 views

Rings of algebraic integers as quotients of polynomial rings

The ring of integers $\mathcal{O}_K$ of a number field $K$ is always isomorphic to some ring of the form $\mathbb{Z}[x_1, ..., x_r]/\mathfrak{p}$, where $\mathfrak{p} \subset \mathbb{Z}[x_1, ..., ...
1
vote
0answers
416 views

Submodule embeddable in a finitely generated module

I have a terminology question. For a commutative ring $A$ (not necessarily Noetherian), $A$-modules that are isomorphic to an $A$-submodule of a finitely generated $A$-module form a fairly good class ...
1
vote
0answers
76 views

Recovering fractional ideals from ideals

Let $R$ be a Noetherian domain; let ${I_{(i)}}$ be a set of fractional ideals in $K$, the fraction field of $R$, indexed by a lattice such that $I_{(i)} I_{(j)} = I_{(i + j)}$. Let $J_{(i)} = I_{(i)} ...
1
vote
0answers
65 views

Irreducibility of a certain matrix variety

Let $R=\mathbb{Q}[x_{i,j}\,:\, 1\leq i,j\leq n]$. Let $M$ be the $n\times n$ matrix $(x_{i.j})$. Let $\chi(M)$ be the characteristic polynomial of $M$. Finally, let $I$ be the ideal of $R$ ...
1
vote
0answers
42 views

Graded Betti Numbers of a Stable Monomial Ideal

Exercise 8.8 in Monomial Ideals by Herzog and Hibi: Let $I\subset S=K[x_{1},...,x_{n}]$ be a stable monomial ideal with $G(I)=\{u_{1},...,u_{m}\}$ and such that for $i<j$, either ...
1
vote
0answers
210 views

when is a sum of idempotents idempotent in commutative ring theory?

As this question demonstrates that the sum of idempotents is idempotent iff every pairwise product is zero, for finite matrices with complex entries. What additional restrictions do we need to put ...
1
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0answers
227 views

separability of commutative rings

Before discussing on the main Question I should recall two notions in the area of commutative rings. By $Max(R)$, we mean the set of all maximal ideals of the commutative unitary ring $R$. ...
1
vote
0answers
219 views

Sums of Strongly z-ideals

In the rings of continuous functions,i.e.$(C(X))$ an ideal $I$ is called strongly $z$-ideal if it is an intersection of some maximal ideals of $C(X)$. i.e. $$I=\cap_{\alpha \in A} ...
1
vote
0answers
280 views

Completion of commutative rings.

Assume that $(R,\mathfrak{m})$ is a commutative local ring of equal characteristic zero. So $R$ contains the field of rationals. The well known $\mathfrak{m}$-adic completion of $R$ provides a ...
1
vote
0answers
199 views

level of rings and stable range of rings

The level $s(A)$ of a ring $A$ with unity $1$ is the smallest natural number $s$ such that $-1$ is a sum of $s$ squares in $A$. (If $-1$ is not a sum of squares in $A$, we say that $s(A) =\infty$.). ...
1
vote
0answers
80 views

Explicit expression for multivariable meromorphic series

Let $K$ be a field. For $m>1$, elements in the ring of power series in $m$ variables over $K$, i.e., $K[[X_1, \cdots, X_m]]$ look like - $$ \displaystyle\sum_{(i_1, \cdots, i_m)\in (\mathbb ...
1
vote
0answers
307 views

Functoriality of a standard integral domain construction.

The evident forgetful functor from fields to integral domains has a left adjoint, namely the construction of the quotient field for a given integral domain. Another standard construction is taking the ...
1
vote
0answers
483 views

nilpotent matrices over polynomial rings

I am looking for an analogue of the Jordan normal form for nilpotent matrices over the polynomial ring ${\mathbb Z}[x_1, \dots, x_n]$. More precisely, is there a description for the orbits of action ...
0
votes
0answers
74 views

Residual Intersections of a complete intersection

Let $R$ be a Cohen-Macaulay local ring and $I=(b_1,\dots,b_s)$ be a complete intersection generated by a regular sequence $\underline{b}$. Let $\mathfrak{a}\subseteq I$ such that ...
0
votes
0answers
54 views

Projective dimension of a sub-ideal

Let $\mathbf{k}$ be a field, and let $S=\mathbf{k}[x_1,x_2,\ldots,x_n]$. Let $I\subset J$ be finitely generated monomial ideals in $S$. Is it true that the projective dimension of $I$ is either ...
0
votes
0answers
32 views

Bigraded analogue of Ratliff-Rush closure filtration

Consider the filtration $\lbrace{I^rJ^s}\rbrace_{r,s\in\mathbb{Z}}.$ What will be the bigraded analogue of Ratliff-Rush closure filtration $\tilde{{I}^n}=\cup_{k\geq1}({I}^{n+k}:{I}^k)$? Will it be ...
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0answers
54 views

Depth of multigraded modules

Can any one please give me some references on depth of multigraded module over a standard multigraded ring.
0
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0answers
89 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 ...
0
votes
0answers
69 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 ...
0
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0answers
120 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 ...
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0answers
70 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!
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0answers
56 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 ...
0
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0answers
283 views

Definitions for Oddness

In the thread Even Xor Odd Infinities I defined odd models of Modular Arithmetic (MA) as models satisfying the axioms of MA and two first order statements. Even XOR Odd Infinities? $O1) \forall x(x=0 ...