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9
votes
0answers
145 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 ...
9
votes
0answers
200 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
votes
0answers
327 views

Can every commutative ring of characteristic $p\in\mathbb P$ be written as the form $R/(p)$ with $R$ being a ring of characteristic $0$?

All rings here are associative, commutative and unital. By a ring of characteristic zero (resp. of characteristic $p$, for prime $p$) I mean a ring $A$ such that the canonical homomorphism $\mathbb ...
6
votes
0answers
89 views

Relative variants of the Jacobson radical

Let $B$ be a commutative ring (with 1). The Jacobson radical can be defined as $$ J(B) = \{b \in B \mid \forall a \in B \colon \quad 1 + a\cdot b \text{ is a unit in } B \} $$ or $$ J(B) =\{ b \in B ...
6
votes
0answers
140 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, ...
6
votes
0answers
170 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
64 views

Result of Larsen and Lunts on rationality of power series with coefficients in a free abelian group

Let $G$ be a free abelian group (not necessarily finitely-generated) and $F$ be the fraction field of the group ring of $G$. Let $\Theta$ be the set of power series in $F[[t]]$ such that each nonzero ...
5
votes
0answers
123 views

On factorization algorithms for $\mathcal{O}[x]$

We know that $\mathsf{LLL}$ algorithm provides factorization procedure that runs in poly time for polynomials in $\Bbb Z[x]$ that are primitive. What other rings $\mathcal{O}$ can we use instead of ...
5
votes
0answers
358 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
250 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
1k 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
114 views

Reference request for $R$-index

Let $R$ be a noetherian domain with field of fractions $F$, let $V$ be a finite-dimensional $F$-vector space, and let $M,N \subseteq V$ be $R$-lattices in $V$ (finitely generated $R$-submodules of $V$ ...
4
votes
0answers
255 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
717 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
220 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
94 views

When a ring is a polynomial ring?

In the paper (2.11) the authors show that if $k^*$ is a separable algebraic extension of $k$ and $x_1,x_2, \ldots, x_n$ are indeterminates over $k^*$ and a normal one dimensional ring $A$ with $k ...
3
votes
0answers
112 views

Integer-matrix representation of a commutative ring

Consider a commutative ring $x_ix_j = N_{ij}^k x_k$, where $N_{ij}^k \in\{0,1,2,3,\cdots\}$, and $\{x_i\}$ is a finite set. (This is actually a fusion ring and $x_i$ are simple objects.) How to find ...
3
votes
0answers
105 views

rings with 'flat functions'

Let $(R,\mathfrak{m})$ be a local ring over a field. Suppose the ring has flat elements, i.e. $\mathfrak{m}^\infty\neq\{0\}$. (The prototype is of course $C^\infty(\Bbb{R}^p,0)$, or a quotient of it, ...
3
votes
0answers
133 views

Annihilators of elements in symmetric algebras

Let $M$ be a module over a commutative ring, and $S(M)$ its symmetric algebra. What elements of $S(M)$ annihilate a given element $m\in M$ ? ($M$ is considered as a submodule of $S(M)$.)
3
votes
0answers
182 views

going down theorem

Typical maps that satisfy "going down theorem" are flat morphisms and integral extensions of normal rings that are integral. Let $Spec(B)\rightarrow Spec(A)$ be a finite type morphism of k-noetherian ...
3
votes
0answers
96 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
149 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
569 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
124 views

When is the torsion submodule a direct factor?

Let $\mathbb{F}$ be a field (of characteristic 0, if needed) and $\mathbf{V}$ an $\mathbb{F}$-vector space. Let $T\in\mbox{End}_\mathbb{F}(\mathbf{V})$ be an endomorphism, and let (following Bourbaki) ...
2
votes
0answers
54 views

Analogue of Bass's Lemma 2.4 on when inverse images of free modules are free

Let $R$ be a Noetherian integral domain. Let $x\in R$ be a prime element. Let $\overline{R}=R/Rx$. Let $P$ be a finitely-generated projective $R$-module. Assume that $\frac{P}{xP}$ is a free ...
2
votes
0answers
67 views

Torsion ideal in symmetric algebra

Let D be a a commutative domain, M be a D-module without torsion and S(M) its symmetric algebra. Is the D-torsion ideal of S(M) the prime ideal of S(M) ?
2
votes
0answers
69 views

Non-existence of commutative rings with many nilpotent elements with some restrictions where matrix powers are efficient

At the moment can't find better reference than "Cycle Enumeration using Nilpotent Adjacency Matrices with Algorithm Runtime Comparisons" though certainly there are others. Consider the following ...
2
votes
0answers
80 views

completion of non-finitely generated ideal

Let consider $A=k[x_{1},x_{2}...]$, the polynomial ring with countably many indeterminates. Then we can consider the completion ...
2
votes
0answers
79 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
175 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
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
0answers
99 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
292 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
547 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
143 views

Structure theorem for infinitely generated modules over a PID

This is a refined version of a question I asked days ago and have no answers yet. I am completely illietarte in algebra so, please, don't kill but explain. The question is all in the title: is there ...
1
vote
0answers
46 views

Modification of nonfree locus

Let $ R $ be a commutative noetherian ring with identity. Let $ M $ be an $ R $-module. By definition the nonfree locus $ NF(M) $ of $ M $ is defined as the set of prime ideals $ {\mathfrak p} $ of $ ...
1
vote
0answers
68 views

Cyclic decomposition of an infinitely generated module

My knowledge of algebra is undergraduate linear algebra, so I apologize for my complete ignorance. Thinking about Jordan normal forms, I unintensionally came to an idea that turned out to be called a ...
1
vote
0answers
60 views

A property of minimal prime ideals in rings with finite chromatic number

Let $R$ be a commutative ring with identity. There are so many ways to associate a graph to $R$. Consider this: take the elements of $R$ (All elements including zero) as vertices an two distinct ...
1
vote
0answers
68 views

Commutative noetherian domains with large fixed rings

Let $R$ be a commutative domain and let $\theta$ be a ring automorphism of $R$. The fixed ring of $\theta$ is defined by $R^{\theta}:=\{r \in R: \ \theta(r)=r \}.$ An ideal $I$ of $R$ is called ...
1
vote
0answers
73 views

Grobner basis for a general algebra

Let $R$ be a quotient of the polynomial ring $\mathbb{C}[x_1,\dots , x_n]$. We fix a $\mathbb{C}^*$ action on $R$ which preserve homogenous components and the multiplication. (The geometric analogue ...
1
vote
0answers
137 views

A family of maximal ideals

Let $m_i $, $i \in I,$ be an infinite family of maximal ideals in a commutative ring with identity (it is not supposed to be Noetherian). When does there exist $j \in I$ such that $\cap_{i\not= j} ...
1
vote
0answers
37 views

Upper bound for the minimum number of generators of the canonical module

Let $P=k[x_1...x_n]$ be a poly over a field. Suppose that $R=k[x_1...x_n]/I$. The canonical module of $R$ is $\omega_R=Ext^{n-dim(R)}_P(R,P)$. The question is that is there any upper bound for the ...
1
vote
0answers
104 views

cohomlogy of Diagonal ring

Let $S=\bigoplus_{\underline n\in\mathbb N^r } S_{\underline n}$ be a standard multigraded ring over a local ring and M be a finitely generated $\mathbb N^r $-graded $S$-module. Let ...
1
vote
0answers
159 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 ...
1
vote
0answers
144 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
75 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
72 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 ...
1
vote
0answers
218 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
466 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
85 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)} ...