The tag has no wiki summary.

learn more… | top users | synonyms

1
vote
0answers
81 views

local rings with finite type maximal ideal

Let $A$ be a local ring with a maximal ideal $\mathfrak{m}$ finitely generated (not principal). Is there a sufficient condition for $A$ to be noetherian? For example, we know that the completion ...
5
votes
0answers
78 views

flatness and derived completion

Let $A$ be a local ring of maximal ideal $\mathfrak{m}$. Let $\hat{A}$ be its completion. If $A$ is noetherian , then we know that $A\rightarrow\hat{A}$ is faithfully flat. If $A$ is not noetherian, ...
0
votes
1answer
97 views

if $R$ is Noetherian local with a finite module of finite injective dimension and if “?” , then $R$ is *Gorenstein*

I know that if $R$ is Noetherian local with a finite module of finite injective dimension, then $R$ is Cohen-Macaulay. Can one add assumptions on $M$, so that $R$ be Gorenstein or Complete ...
0
votes
0answers
29 views

normality of truncated arc space

Let $X=Spec(A)$, with $A$ a normal $k$-algebra of finite type, $k$ is a field. For any integer $n$, let $X(k[t]/(t^{n}))$ the $n$-th truncated arc space, is it also normal? Same question for ...
3
votes
3answers
150 views

canonical module can be identified with an ideal. how can one reach that ideal?

Let $R[[X,Y,Z]]/(X,Y)\cap (Y,Z)\cap(X,Z)$. then $R$ is Cohen-Macaulay ring and has a canonical module, $K$. By Proposition 3.3.18 of Bruns_Herzog, $K$ can be identified with an ideal in $R$. So we ...
8
votes
0answers
85 views

Weierstrass division theorem for henselian rings

Let $A$ be an henselian local noetherian ring. There is an old result of Lafon ("Anneaux henséliens et théorème de préparation" (1967)), which says that if $A$ is analytically normal and of ...
1
vote
1answer
104 views

Graded version of Baer's Criterion

Baer's Criterion for injectiveness of modules says: "An $R$-module $E$ is injective iff for all ideals $I$ of $R$, every homomorphism $f\colon I \to E$ can be extended to $R$." I wonder if there is a ...
-2
votes
0answers
168 views

intuitive interpretation of the multiplicity and a reference [migrated]

Although logically i can understand and use multiplicity, yet, the concept of multiplicity of a module is not completely clear for me. I wonder what idea is behind this definition and What is ...
0
votes
0answers
75 views

: References on complete intersections rings

As we know, for the local ring, we have regular $\subset$ complete intersection $\subset$ Gorenstein $\subset$ Cohen-Macaulay but it seems that the refenences of complete intersection are more ...
3
votes
1answer
480 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
210 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 ...
4
votes
1answer
125 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
1answer
91 views

Matching power series to infinity

As pointed out by Makoto, on this question about power series rings and the axiom of choice, an idea I had needed the axiom of dependent choice to work. However, the construction raises another ...
2
votes
1answer
184 views

finiteness dimension

$R$ is a local Noetherian ring. $f_I(M)$, the finiteness dimension of a module $M$ relative to $I$, is defined in ...
5
votes
2answers
873 views

Is being finitely generated a local property?

I am trying ot figure out a proof of the following fact, that I believe is true, but it seems to me that something is lacking. Suppose we have commutative, unitary rings $A,B$ and a (unit preserving) ...
1
vote
1answer
147 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 $S$?
1
vote
1answer
230 views

intuitive interpretation of analytic spread

I am studying analytic spreads from Bruns-Herzog's book. The definition is clear but calculation of the analytic spread of an ideal is hard for me in practice. I wonder if it is hard for you too. ...
1
vote
0answers
79 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
votes
1answer
133 views

if $ \lambda (I)= \dim R$, can one claim that $I$ is an $m$-primary ideal?

definition from Bruns-Herzog: It is easy to see that if $I$ is a $m$-primary ideal of $R$ then $ \lambda (I)= \dim R$. I wonder if the converse is true: if $ \lambda (I)= \dim R$, can one ...
1
vote
0answers
136 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 $I$⊂$P$?
2
votes
1answer
179 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
2answers
475 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 ...
5
votes
2answers
216 views

Integral domains with totally ordered spectra

In my research I ended up trying to prove some properties of integral domains such that their spectrum is a totally ordered poset. Are there some nice (ubiqitous/natural) examples of such domains, ...
0
votes
0answers
89 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
1answer
70 views

Bounded Index of Nilpotency of $R[x]$

A ring $R$ is called with bounded index (of nilpotency) $n$ if $n$ is the smallest natural number such that $a^n=0$ for all nilpotent $a \in R$. Now let $R$ be a commutatitve ring with bounded index ...
1
vote
0answers
49 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.
32
votes
7answers
3k views

Does $SL_3(R)$ embed in $SL_2(R)$?

Is there any non-trivial ring such that $SL_{3}(R)$ is isomorphic to a subgroup of $SL_{2}(R)$? $SL_{3}(\mathbb{Z})$ is not an amalgam, and has the wrong number of order $2$ elements to be a subgroup ...
0
votes
1answer
207 views

reduction of an admissible filtration

Let $(R,m)$ be a local ring and $I$ an $m$-primary ideal of $R.$ $\lbrace I_n\rbrace_{n\in\mathbb{Z} }$ is called $I$-admissible filtration 1) if $m\geq n$ then $I_m\subset I_n.$ 2) for all $m,n,$ ...
1
vote
0answers
144 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
1answer
177 views

Quotients and radicals

Let $I, J$ be ideals in a commutative ring with identity $R$. Define the quotient ideal $(I : J)$ by $$(I : J)=\{x\in R : xJ\subseteq I\}.$$ Define the radical $r(A)$, of an ideal $A$ of $R$ by ...
0
votes
0answers
59 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 ...
5
votes
1answer
230 views

Does $ \text{mult}(R / I) = d_{1} \cdots d_{r} $ imply that $ (f_{1},\ldots,f_{r}) $ is an $ R $-regular sequence?

We define the multiplicity of an $ R $-module $ M $ of dimension $ d > 0 $ to be $$ \text{mult}(M) \stackrel{\text{df}}{=} \text{LC}(P_{M}) \cdot (d - 1)!, $$ where $ P_{M} $ denotes the Hilbert ...
0
votes
1answer
90 views

$I$ is principal if and only if $ID_M$ is principal for all $M \in Max(D)$

In a semmi-quasi local domain $D$ ( i.e. $D$ has finitely many maximal ideals), an ideal $I$ is principal if and only if $ID_M$ is principal for all $M \in Max(D)$. [See Comm. Rings by Kaplansky, ...
42
votes
1answer
2k views

A condition that implies commutativity

Let $R$ be a ring. A notable theorem of N. Jacobson states that if the identity $x^{n}=x$ holds for every $x \in R$ and a fixed $n \geq 2$ then $R$ is a commutative ring. The proof of the result for ...
1
vote
0answers
430 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 ...
28
votes
3answers
2k views

Even XOR Odd Infinities?

Modular Arithmetic (MA) has the same axioms as first order Peano Arithmetic (PA) except $\forall x (Sx \ne 0)$ is replaced with $\exists x(Sx = 0)$. ...
-3
votes
1answer
110 views

Isomorphic quotient of a Module over Noetherian commutative algebra [closed]

I have a nice solution to the following problem and I thought of writing a paper about it but beforehand, I wanted to ask the problem here to see if this is an easy problem and if you people can solve ...
1
vote
0answers
79 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
1answer
119 views

R is regular local rings of Krull dimension 2.Can we find any ideal of height 2 different from m? [closed]

Maybe it is too easy but I want to know that: If $R$ is regular local ring of Krull dimension $2$ and $m$ is the maximal ideal of $R$. (It means that height $m$ is $2$). Can we find any ideal of ...
11
votes
7answers
722 views

Properties of rings that have an elegant description in terms of the associated category of modules

Suppose $A$ is a ring. Then $A$ happens to be a division ring iff every left $A$-module is free. (See here for proofs). I think this is very beautiful; what other properties of rings have an elegant ...
7
votes
2answers
336 views

What is the probability that a random sequence of polynomials is regular?

Let $k$ be a finite field or a field with a height function, such as a number field. Consider the ring $k[[x_1,\dots, x_n]]$ and let $\mathfrak{m}$ be its maximal ideal. What is the asymptotic ...
2
votes
0answers
71 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 ...
0
votes
1answer
115 views

question about valuation ring

$k$ algebraically field, $A$ $k$ algebra and valuation ring of $K$ ($K$ field fraction of $A$) and we have the transcendence degree of $K$ over $k$ is one. i want to ask if $A$ is noetherian ring?
1
vote
0answers
67 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
2answers
286 views

In this special situation, does $M \otimes B=0$ imply $M=0$?

Let $\Phi:A \rightarrow B$ be a flat morphism of commutative rings. Let $f \in A$, not a unit and $A/fA \cong B/fB$ induced by $\Phi$. Let $M$ be an $A_f$-module. Is it true that $M \otimes_A B = 0 ...
2
votes
1answer
130 views

Nilradical and Newton's identities

Let $R$ be a commutative ring with unity such that $n!$ is not a zero-divisor. Let $s_1=\sigma_1,s_2,s_3\cdots$ and $\sigma_1,\sigma_2\cdots$, (convention: if $k>n$, then $\sigma_k=0$) be elements ...
2
votes
1answer
217 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 ...
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 ...
5
votes
1answer
179 views

Units of $\mathbf Z[X,Y]/(P(X,Y))$

Let $P(X,Y)\in \mathbf Z[X,Y]$ be an irreducible polynomial and let $A$ denote the quotient ring $\mathbf Z[X,Y]/(P)$. What is known about the group of units of $A$? It's not even clear to me that ...
3
votes
1answer
136 views

Idempotent fractional ideals of a Noetherian domain

Let $R$ be a commutative Noetherian domain, $K$ its fraction field, and $J$ a fractional ideal (i.e. a finitely generated sub-$R$-module of $K$) such that $J^2=J$. Is it true that $J=0$ or $J=R$? If ...