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2
votes
1answer
149 views

Whether the result that an ultraproduct which satisfies ACCP is automatically a field generalizes to ultrafilters on larger indexing sets

Let $F$ be an ultrafilter on some set $X$, $R$ an integral domain and $R_F$ the resulting ultraproduct ring. For an element $(a)$ in the product ring of $R$ indexed by $X$, denote its equivalence ...
1
vote
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
1answer
269 views

The set of Upper semi-continuous functions as a ring.

I should recall that the surgenfery topology on the real numbers is denoted by $\mathbb{R}_l$, and has the set {$[a , b): a,b \in \mathbb{R} $} as it's base. If $X$ is a topological space, an upper ...
1
vote
1answer
376 views

Lifting identities of formal power series

I am looking for a possibly general class of algebraic structures (maybe special topological rings) in which one can deduce identities of concrete power series from formal ones. This class should ...
6
votes
3answers
986 views

functional subrings

I should recall the notion of maximal subring of a commutative unitary ring $R$. Def: A commutative ring $S$ is called a maximal subring of $R$ if $S \subset R$ and if $T \subset R$ constitute a ...
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} ...
7
votes
4answers
641 views

Quotient rings of $C(X)$

Let $X$ be a Tychonoff topological space. Consider the ring $C(X)$ of all continuous real valued functions on $X$. For what conditions on an ideal $I$ of $C(X)$, we could deduce that the quotient ring ...
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 ...
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 ...
5
votes
1answer
412 views

Divisibility and factorization in rings that are not integral domains

In my course notes for an undergraduate course "Algebra I", I wrote at the point when I'm introducing the notion of divisibility in rings (in a section on unique factorization): We want to study ...
0
votes
1answer
98 views

Uniqueness of Hensel factors of a polynomial (invariant to change of “basepoint”)?

An important component of algorithms for factoring multivariate polynomials over a commutative ring $R$ is Hensel lifting. Here's a brief, concrete example to set the stage for my question: Let $f ...
2
votes
1answer
279 views

Frobenius base change of etale maps

Let $A$ be a characteristic $p>0$ commutative ring. Let $B$ be a finitely presented etale $A$ algebra i.e. $$ B=A[x_1,\ldots,x_n]/(f_1,\ldots,f_n), $$ with $det(\frac{\partial f_i}{\partial ...
2
votes
2answers
442 views

Infinite domain with finite number of prime ideals(elements)

While trying to prove one property of commutative rings with units I can't prove one fact without assuming existence of infinitely many different prime ideals or elements. I tried to test if it was ...
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 ...
8
votes
1answer
2k views

Maximal ideals of Z[x,y]

we know that the maximal ideals of ${\mathbb Z}[x]$ are of the form $(p, f(x))$ where $p$ is a prime number and $f(x)$ is a polynomial in ${\mathbb Z}[x]$ which is irreducible modulo $p$. Is it true ...
1
vote
1answer
200 views

semilocal total quotient ring whose J(R) is not zero

I am interested in rings in which every non-unit is a zero divisor. Can you give me an example of such a ring that also has FINITELY many maximal ideals (semilocal), and whose Jacobson radical is not ...
1
vote
0answers
200 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
1answer
161 views

relation between Min(R) and Min(R^)

Let $\hat{R}$ is $m$-adic completion of a local ring $(R,m)$.What is the relation between $Min R$ and $Min \hat{R}$. we know that $\hat{R}$ is faithfully flat $R$-module. $Min R$=set of all minimal ...
9
votes
1answer
801 views

Lengths over a local ring

Let $A$ be a noetherian domain, $\mathfrak{m}$ а maximal ideal, $s$ a non-zero element of $\mathfrak{m}$, $d= \dim A_\mathfrak{m}$. Is the following claim true? Claim: For any $\epsilon>0$, there ...
2
votes
1answer
447 views

relation between Ass Ext(M,N) and Ass M ,Ass N

Let R be a noetherian ring and M , N be finitely generated $R$-modules. Then what is the relation between $Ass Ext^i_R(M,N)$ and $Ass M, Ass N$? $Ass$ means set of associated prime ideals. It's well ...
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 ...
3
votes
2answers
706 views

lim Ext(a^n/b^n,R)=0

Is it true that: Let $R$ be a local ring and $\dim R= d$. If $b\subset a$ be two proper ideals of $R$ then for $ n\in {\Bbb N}$, $\varinjlim Ext^d_R(a^n/b^n,R)=0$
3
votes
1answer
699 views

Serre condition $(S_n)$

We know that a finitely generated $R$-module $M$ satisfies the $(S_n)$ condition if $$\operatorname{depth} M_p \geq \min(n,\dim M_p)$$ for every $p\in \operatorname{Spec}R$. It's well known that ...
0
votes
1answer
441 views

associated prime ideal [duplicate]

Possible Duplicate: minimal prime devisor(MinAss R) Hello All,is This conclusion true? $(R,m)$ be a local ring.if every associated prime ideal of $R$ be minimal then every associated prime ...
2
votes
1answer
453 views

Minimal prime divisors (MinAss R)

Hello All,is This conclusion true? If $(R,m)$ is a local ring and $ Min Ass R=Ass R$ then can we conclude that $Min Ass \hat{R}=Ass \hat{R}$? ($\hat{R}$ is $m$-adic completion of $R$) ...
0
votes
1answer
227 views

Chain of ideals in a complex algebra

Suppose $\mathfrak{A}$ is an unital algebra over complex numbers and $\mathfrak{J}$ is chain of left-ideals in $\mathfrak{A}$ ordered by inclusion such that none of its elements is countably ...
10
votes
1answer
773 views

Affine scheme on spec(A) of a ring A as the sheafification of a pre-sheave on spec(A)?

It is obvious that there is a parallel between the definition of structure sheaf of $\operatorname{Spec}(A)$ versus the sheafification of a pre-sheaf. The definition of the sheaf $\mathscr F^+$ ...
2
votes
1answer
279 views

Finiteness of normalization of Noetherian normal domain

I have the following question: Let $A$ be an integrally closed Noetherian domain, $K$ its field of fractions. let $L$ be a finite extension of $K$, and $B$ the integral closure of $A$ inside $L$. Is ...
0
votes
1answer
219 views

Embedding commutative associative rings in non associative rings [closed]

Let $R$ be a commutative and associative ring with unit. Can $R$ be embedded in a ring $\hat{R}$ wich is both non commutative and non associative ? Thanks guys !
1
vote
2answers
301 views

quotient of integral polynomials not being integral

So let $R$ be an integral domain and $K$ its fraction field. Let $f,g,h\in K[x]$ be 3 monic polynomials such that $f=gh$. So I would like to have a simple example of a ring $R$ for which one has that ...
10
votes
1answer
448 views

Are there non-reflexive modules isomorphic to their bi-dual?

Let $M$ be an $R$-module. We say that $M$ is reflexive if the natural map $M\rightarrow M^{**}$ is an isomorphism. I'd like to know if there exists a module isomorphic to its bi-dual but not ...
6
votes
2answers
2k views

Periodic matrices

A square matrix $M$ such that $M^{k+1}=M$, for some positive integer $k$, is called a periodic matrix. Can we characterize the periodic matrices in $\mathcal{M}_n(\mathbb{Z})$? If we replace ...
2
votes
2answers
279 views

Can normalisations of curves over a perfect field change residue fields?

Does anyone know an example of a curve $X$ over a perfect field $k$ such that if $\tilde{X}$ is its noramlisation, there exists a point $x \in X$ and a point $y \in \tilde{X}$ over $x$ such that $k(y) ...
8
votes
2answers
558 views

Is the support of an Artinian module finite?

Let $R$ be a commutative Noetherian ring, $M$ is an Artinian $R$-module. Is the set $Supp_R(M)$ finite? Thanks.
7
votes
2answers
700 views

Modules over Laurent series rings

Let $k[x]$ be the ring of polynomials over a field k in one variable x. A $k[x]$-module is a k-vector space together with a linear endomorphism (the action of x). The field $k(x)$ of rational ...
13
votes
2answers
951 views

Dimension 1 prime ideals in the intersection of two maximal ideals

This question/problem really comes from a fact in algebraic geometry, where it says that given an irreducible variety $V$ ($\dim V \geq 2$) then for any given pair of points $x,y\in V$ there is an ...
1
vote
2answers
676 views

Rational power series

If we let $R=\mathbb{Z}[x]$ and $D=\mathbb{Z}[[x]]$. We say that $z\in D$ is rational if there is $g\in R$, $g\ne 0$ such that $zg\in R$. Let $S$ be the set of all rational elements in $D$. Then $S$ ...
1
vote
2answers
156 views

Is there a relationship between the right global dimensions of R and R[1/v]?

A few days ago I asked a similar question about Krull dimension and got fantastic answers. Unfortunately, for the application I have in mind (a question on ring spectra), Krull dimension doesn't ...
30
votes
3answers
3k views

What the heck is the Continuum Hypothesis doing in Weibel's Homological Algebra?

On page 98 of Weibel's An Introduction to Homological Algebra he mentions that the ring $R = \prod_{i=1}^\infty \mathbb{C}$ has global dimension $\geq 2$ with equality iff the continuum hypothesis ...
13
votes
4answers
2k views

Is there a Galois correspondence for ring extensions?

Given an ring extension of a (commutative with unit) ring, Is it possible to give a "good" notion of "degree of the extension"?. By "good", I am thinking in a degree which allow us, for instance, to ...
6
votes
2answers
809 views

Is every poset the poset of prime ideals of a ring?

The answer to this question, as it is, is trivially false, for one necessary condition is the existence of maximal element(s), i.e., maximal ideals exist and are prime. My question was inspired from ...
6
votes
4answers
2k views

Applications of commutative algebra

Hi. I'm preparing a thesis in commutative algebra, and when I say this to my friends they always ask me what are the applications to "real-world", and I don't know what to answer. This let me think ...
1
vote
1answer
283 views

A problem on Moebius transformations

We have the following result: Let $R=\mathbb{C}[t]_f$, with $f=(t-a_1)(t-a_2)\cdots (t-a_n)$. Then the automorphism group of $R$ is isomorphic to the group of all Moebius transformations which fix ...
2
votes
2answers
601 views

A problem for finite dimensional commutative algebra

Let $(A,m)$ be a local commutative associative algebra over the field of complex numbers, $m^n\ne 0$, $m^{n+1}=0$ for some $n>0$, and (1) $A$ is finite dimensional as vector space (2) for any ...
10
votes
4answers
853 views

Are units of rings of functions on algebraic varieties finitely generated (mod. constants)?

Hello, Consider the following question. Let $A$ be a finitely generated reduced algebra over an algebraically closed field $k$. Consider the group of units of $A$, modulo the group $k^*$. Is this ...
18
votes
3answers
1k views

Invariance of $Z[x]$ under a self-equivalence of the category of commutative rings with 1.

Let $\mbox{Rings}$ be the category of commutative rings with $1$. Is there an equivalence of categories $F: \mbox{Rings} \to \mbox{Rings}$ s.t. $$F(\mathbb{Z}[x])\not\cong \mathbb{Z}[x]?$$
4
votes
1answer
553 views

module of differentials of formal power series ring and of its field of quotiens

For any $A$-algebra $B$ ( commutative ring with 1 ), we have the existence of $\Omega_{B/A}$, the module of relative differentials of $B$ over $A$, which can be defined by an universal property. In ...
1
vote
1answer
359 views

Unimodular column property

Hi, I know that if $R$ is a ring such that every projective $R$-module finitely generated is free then $R$ has the unimodular column property. I would like to know if there is a ring $R$ that doesn't ...
7
votes
1answer
677 views

Direct sum of injective modules over non-Noetherian rings

Hi. I know, by the Bass-Papp theorem, that if every direct sum of injective $R$-modules is injective then $R$ is Noetherian. I would like to know if there exists a direct sum of injective $R$-modules ...
1
vote
3answers
507 views

Stably free module not finitely generated is free

Hi. I have read that stably free modules not finitely generated are free; this is proved in M.R. Gabel, stably free projectives over commutative rings, Thesis, Brandeis Univ., Waltham, MA 1972. But ...