All Questions
Tagged with local-rings ac.commutative-algebra
171 questions
1
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1
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264
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Flat and algebraic (non-integral) local rings extension $R \subseteq S$ with $m_RS=m_S$
Let $R \subseteq S$ be two Noetherian local rings, not necessarily regular, which are integral domains,
with $m_RS=m_S$, namely, the ideal in $S$ generated by $m_R$ (= the maximal ideal of $R$) is $...
- 2,837
0
votes
1
answer
248
views
Given a unitary commutative ring $R$, what are the rings $R\langle x,y\rangle/(x^2-A,y^2-B,yx-a-bx-cy-dxy)$ called
We are studying the rings
$$
R \langle x, \, y \rangle\,\big/\left(x^2-A, \, y^2-B, \, yx-a-bx-cy-dxy \right)
$$
Do you know if they have a name?
1
vote
1
answer
147
views
Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property?
Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property?
The examples of rings not isomorphic to their opposite that I know of are not ...
5
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1
answer
395
views
Kähler differentials on an Artinian local ring
Suppose $R$ is a commutative Artinian local ring over an algebraically closed characteristic 0 field $k$. Suppose $f\in R$ is such that $df=0$ (in the sense that the element $df$ vanishes in the ...
- 2,834
4
votes
0
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265
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Sections of smooth morphisms over henselian rings
Let $(A,\mathfrak m)$ be a henselian local ring. Let $R$ and $S$ be $A$-algebras of finite type and $f\colon R\to S$ be a smooth morphism. Assume that the induced morphism $R/\mathfrak m R\to S/\...
- 1,590
6
votes
1
answer
499
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Do you know which is the minimal local ring that is not isomorphic to its opposite?
The most popular examples are non-local rings and minimal has 16 elements. I am interested in knowing examples of local rings not isomorphic to their opposite.
1
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0
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94
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What would be the quotient groups $U_{\mathrm{gen}}/U_{\mathrm{gen}}^{(n)}$ and $U_{\mathrm{gen}}^{(n)}/U_{\mathrm{gen}}^{(n+1)}$?
Let $K \supseteq \mathbb{Q}_p$ be a $p$-adic field with ring of integer $O$ and maximal ideal $m$. Let $O^*$ be the group of units in $O$.
Consider the group of units $U^{(0)}=U=O^*$ and $U^{(n)}=1+m^...
- 930
9
votes
2
answers
815
views
Ideal norm in orders
Let $\overline{T}$ be a Dedekind ring such that $\overline{T}/\overline{I}$ is finite for every nonzero ideal $\overline{I}$ of $\overline{T}$. Let $T$ be a subring of $\overline{T}$ with the same ...
- 328
8
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0
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221
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Finitely generated commutative rings with the same profinite completion
Let $R_1$ and $R_2$ be two finitely generated commutative rings. Assume that their profinite completions are isomorphic: $\widehat{R_1}\cong \widehat{R_2}$.
Suppose that $R_1$ is a domain. Does ...
- 1,418
6
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2
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421
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Does the category of local rings with residue field $F$ have an initial object?
Let $F$ be a field. Does the category $C_F$ of local rings $R$ equipped with a surjective morphism $R\longrightarrow F$ have an initial object?
This is, for instance, true if $F=\mathbb{F}_{p}$ for ...
- 1,805
3
votes
2
answers
338
views
Isomorphism between finite algebras over ${\Bbb Z}_p$
Let $\pi \colon R \twoheadrightarrow {\Bbb T}$ be a surjective ring homomorphism between finite algebras over ${\Bbb Z}_p$. Further, we suppose the following three conditions$\colon$
$R$ is a ...
- 563
4
votes
1
answer
158
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injective hulls in mixed characteristic
Let $R=\underleftarrow\lim (R/\mathfrak m^i)$ be a complete local ring, with residue field $k=R/\mathfrak m$,
and let's assume that $R$ is Noetherian.
If $R$ is a $k$-algebra, then
I believe that ...
- 43.2k
10
votes
2
answers
1k
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Krull dimension of a local ring and completion
Let $A$ be a local ring (not noetherian) of finite Krull dimension such that its maximal ideal $\mathfrak{m}$ is of finite type.
Let $\hat{A}$ be its $\mathfrak{m}$-adic completion.
Do we have that $\...
- 3,472
1
vote
0
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94
views
Generators for Ideals in ring of multivariate Laurent Polynomials
Consider the following problem:
Find an ideal $I \subset \mathbb{Q}[x^{\pm}_1,x^{\pm}_2,x^{\pm}_3]$ such that $I_{aff} \subset \mathbb{Q}[x_1, x_2, x_3] = I \cap k[x_1, x_2, x_3]$ requires more ...
- 11
1
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1
answer
224
views
Power series ring $\Theta[[X_1,\ldots,X_d]]$ and prime ideals
Let $\Theta$ be a domain. We shall choose $d$ elements $\theta_1,\ldots,\theta_d \in \Theta$ such that any chosen $j$ elements $\theta_{i_1},\ldots,\theta_{i_j}$ form a prime ideal $(\theta_{i_1},\...
- 563
0
votes
0
answers
287
views
On the product in the power series ring
Let $A_n \colon= K[[X_1,\ldots,X_n,Y_1,\ldots,Y_n]]$ be a power series ring over a field $K$ in $2n$ variables and ${\frak m}_{A_n}$ be the unique maximal ideal of $A_n$.
Suppose we have two ...
- 563
1
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0
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165
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Structure of Complete Local Rings
Let $X$ be a proper $n$-dimensional $k$-scheme and $x \in X$ a closed point. Consider the stalk $\mathcal{O}_{X,x}$. We consider now it's completion $O_{X,x}^{\wedge}$ wrt it's maximal ideal $m_x$.
...
- 6,006
7
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1
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268
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Additive group of local rings
Is there a theory or characterization for those finite $p$-groups that can be considered as the additive group of a finite local commutative ring with identity?
- 155
3
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1
answer
193
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Complete local rings, automorphisms and approximation
Consider two local morphisms $f,g: B\rightarrow A$ of noetherian complete local rings and $f$ surjective.
Does there exist an integer $n\in\mathbb{N}$, such that if $f=g \mod \mathfrak{m}_{A}^{n}$ ...
- 3,472
1
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0
answers
138
views
Power series ring $R[[X_1,\ldots,X_d]]$ over a domain $R$
Let $R$ be a domain and
\begin{align*}
T \,\colon= R[[X_1,\ldots,X_d]].
\end{align*}
Suppose that we have $d$ elements $f_1,\ldots,f_d \in T$ and let us consider an ideal $J$ of $T$ such that $(f_1,\...
- 563
1
vote
0
answers
310
views
Primes of the power series rings
Let $A_n \colon= K[[X_1,\ldots,X_n]]$ be a $n$-variable formal power series ring. By setting $X_n \mapsto 0$, we obtain a natural surjection
\begin{equation*}
\psi_{n,n-1} \colon A_n \...
- 563
1
vote
0
answers
78
views
Relation between lifts of simple roots and lifts of idempotents (Henselian property)
Let $f:A\to B$ be a morphism of commutative rings. Given a monic $\varphi\in A[x]$ write $Z(A,\varphi)$ for the set of simple roots of $\varphi$ in $A$. Consider the following properties of $f:A\to B$....
- 10.5k
3
votes
1
answer
239
views
commutative ring satisfying descending chain condition on radical ideals
Let $R$ be a commutative ring with unity which satisfies d.c.c. on radical ideals. then does $R$ satisfy a.c.c. on radical ideals ? If this is not true in general, then what happens if we also assume $...
- 1,209
0
votes
1
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339
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Under which conditions on the homogeneous ideal $ I $, the quotient ring $ \mathbb{C} [X_0, \dots, X_n]/I $ is a regular ring?
If $ I $ is a homogeneous ideal of the ring of homogeneous polynomials $ \mathbb {C} [X_0, \dots, X_n] $ , under which conditions on the homogeneous ideal $ I $, and particularly on $ I_m $, the $m$ -...
- 325
8
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1
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257
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Minimal resolution of local cohomology module
Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d\geq 1.$ Suppose $\lambda(H_{\mathfrak m}^i(R))<\infty$ for some $0\leq i\leq d-1.$
Question Can we say anything about Betti numbers ...
- 1,713
-2
votes
2
answers
764
views
Reduced ring with all non-prime ideals finitely generated
Let $R$ be a reduced ring with all non-prime ideals finitely generated. Then is $R$ Noetherian ? If not, then is it true at least in the local case ?
Without reduced assumption, it is not true even ...
user111492
9
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1
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444
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Rings with all non-prime ideals finitely generated
Motivated by this question, I would like to ask:
If all non-prime ideals in a ring are finitely generated, then is the ring Noetherian? Can we at least say anything in the local case?
Note that ...
user111492
5
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1
answer
1k
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local ring all whose non-maximal ideals are finitely generated
Let $(R, \mathfrak m)$ be a commutative local ring such that every non-maximal ideal is finitely generated. Then, is $R$ Noetherian i.e. is $\mathfrak m$ finitely generated ideal ?
It is easy to see ...
user111524
1
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1
answer
465
views
Valuation ring whose maximal ideal and every ideal of finite height are principal
Let $(R, \mathfrak m)$ be a valuation ring such that $\mathfrak m$ and every ideal of finite height is principal. Then is $R$ Noetherian , i.e. a discrete valuation ring ?
user111492
5
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1
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308
views
When is a zero dimensional local ring a chain ring?
A commutative ring with identity is called a chain ring if all its ideals form a chain under inclusion. I want to know is there any proof for the fact that a zero dimensional local ring is a chain ...
4
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1
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348
views
Given a non-field local domain $R$, finding a dominating Valuation ring whose residue field is algebraic/finite extension of the residue field of $R$
Let $(R, \mathfrak m)$ be a non-field local domain with fraction field $Q(R)$ . Let $k_{R}:=R/m$.
We know that there is a Valuation ring $(V,\mathfrak m_V)$ such that $R \subseteq V \subsetneq Q(R)$ ...
user111524
0
votes
0
answers
53
views
a generalization of group (monoid with order-by-order invertible elements)
Fix a filtered monoid, $H=H_0\supsetneq H_1\supsetneq H_2\supsetneq\cdots$. Suppose for any $h\in H$ and any $n\in \Bbb{N}$ exists $h_n\in H$ such that $h\cdot h_n\in H_n$ and $h_n\cdot h\in H_n$. If ...
- 2,232
12
votes
3
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790
views
$K[[X_1,...]]$ is a UFD (Nishimura's Theorem)
Let us define the infinitely-many-variable formal power series ring
$$
K[[X_1,\ldots]] \colon= \underset{m \geq 1}{\varprojlim}\,K[[X_1,\ldots,X_m]].
$$
$K[[X_1,\ldots]]$ is known to be a UFD by a ...
- 563
4
votes
0
answers
177
views
What kind of module is this?
Recall that, if $R$ is a commutative ring, then a suitably finite $R$-module $M$ is projective if and only if the localization $M_\mathfrak{m}$ is a direct sum of finitely many copies of $R_\mathfrak{...
- 985
1
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0
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46
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Integral closure of lexsegment ideal
Let $R=k[x_1,\ldots,x_d]$ where $k$ is a field and $I$ be a lexsegment ideal of $R$ and $l(I)=d$ (where $l(I)$ is analytic spread of $I$).
Is $I$ integrally closed?
If I is generated by elements ...
- 1,713
7
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1
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1k
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indecomposable module over a local ring
I ask this in mathematics for some days.it doesn't have an answer up to now. https://math.stackexchange.com/questions/2565828/indecomposable-module-over-a-local-ring
As we all know, for an arbitrary ...
- 496
9
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2
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713
views
The projective covers of Artinian module
The injective hull for a module always exists, however over certain rings modules may not have projective covers. I have a question.
If $A$ is an Artinian module on a Noetherian local ring $R$ then $...
4
votes
2
answers
353
views
Canonical module of a semigroup ring
Let $S$ be a numerical semigroup and $k[S]$ is the associated semigroup ring. I would like to compute canonical module $\omega$ of $k[S].$
I want to show that $\omega=k[t^{-n}:n\in\mathbb Z\setminus ...
- 1,713
2
votes
1
answer
195
views
Length of a module and Frobenius map
Let $(R,m)$ be a regular local ring of dimension $d$ and char $p>0.$ Let $F^e:R\longrightarrow R$ defined by $r\longrightarrow r^{p^e}$be the Frobenius map.
How to compute $l(R/m^{[p^e]})?.$
I ...
- 1,713
2
votes
0
answers
77
views
Analytic spread of an ideal after reduction
Let $(R,m)$ be a local ring and $I$ an ideal in $R.$ Let $l(I):=\dim \bigoplus_{n\geq 0}(I^n/mI^n)$ and $x\in R\setminus I.$
My question is
what is the relation between $l(I)$ anf $l(I+(x)/(x))?$
- 323
1
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0
answers
294
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Is it true that the functor of completion of a module over a local ring is injective on isomorphism classes?
Let $A$ be a commutative Noetherian local ring and $\hat A$ be its completion. Then we have the functor of completion from the category of finitely generated $A$-modules to the category of finitely ...
- 1,484
1
vote
1
answer
241
views
Localization of a maximal Cohen-Macaulay module
Let $(R,m)$ be a Cohen-Macaulay local ring of dimension $d\geq 2$ and $M$ an module with depth$M=d.$
Is there any example of $M$ such that
$(1)$ $M_p$ is not free for some $p\in Ass(R)$ and
$(2)$...
- 323
0
votes
1
answer
114
views
Adic filtration and integral closure
Let $(R,m)$ be a Noetherian local domain whose integral closure $S$ is too. Also assume that $S$ is module-finite over $R$.
Let $x\in m^k\setminus m^{k+1}$ and $u\in S^\times$ such that $ux \in R$. ...
- 3,079
2
votes
0
answers
221
views
Meaning of the statement "$a\in I$ is a general element of $I$"
Suppose $I$ is an ideal in a Noetherian local ring $(R,m)$. In some papers I have seen the following statement:
"$a\in I$ is a general element of $I$".
What is the definition of general element ...
- 1,713
0
votes
1
answer
73
views
A question on infinite local rings which are not division ring
Is it true that if $(R,m)$ is a (not necessary commutative) local ring then $R$ and $m$ have the same cardinal ? (Exclude TWO trivial cases: when $R$ is finite and when $R$ is a division ring)
On ...
- 271
1
vote
0
answers
285
views
Analytic spread of an ideal
How to calculate analytic spread of the ideal $I=\left<xyw^2,xyz^2,xw^2+yz^2\right>$ in $\mathbb Q[x,y,z,w]?$
I think it is 3.
- 1,713
0
votes
1
answer
260
views
Analytic spread of localization of an ideal
Let $J$ be an ideal in a Noetherian local ring $(R,m)$. It is well known that for any prime ideal $p\in Spec(R)$, $l(J_p)\leq l(J)$, where $l(J)$ is the analytic spread of $J$.
Q) Are there ...
- 1,713
5
votes
1
answer
509
views
how to pass from algebraic power series to the analytic ones
Fix a field of zero characteristic, $k$, e.g. $\Bbb{R}$ or $\Bbb{C}$. Suppose $k$ is normed (and complete for its norm). Consider the ring extensions: $k[x_1,..,x_n]\subset \ k<x_1,..,x_n> \ \...
- 2,232
1
vote
0
answers
112
views
Asymptotic stability of prime divisors
Suppose $I$ is an ideal in a formally equidimensional local ring $R.$ Let $A(I)$ and $\overline A(I)$ denote Ass$R/I^n$ and Ass$R/\overline{I^n}$ for all large $n$ respectively.
My question is
What ...
- 1,713
0
votes
1
answer
300
views
Behaviour of length function under faithfully flat extension
Let $(R,m)$ and $(S,n)$ be local Noetherian rings such that $S$ is a faithfully flat extension of $R$. Let $J\subsetneq I $ ideals of $R$.
Can we relate $l_R(I/J)$ and $l_S(IS/JS)$?
PS: Here $l(-)$ ...
- 1,713