The commutative-rings tag has no usage guidance.

**5**

votes

**1**answer

176 views

### In a noetherian commutative ring with only one associated prime, is the nilradical locally free?

The title says it all.
I suspect that the answer in general is no, although my intuition tells me that a jump in the dimension of the fibre of the nilradical at some point of Spec(A) can occur only ...

**5**

votes

**0**answers

128 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 $\...

**3**

votes

**0**answers

108 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, ...

**2**

votes

**0**answers

73 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 ...

**0**

votes

**2**answers

145 views

### Making idempotent element by a relation [closed]

Let $R$ be a commutative ring with identity and let $a, b \in R$ such that $a=ab$. How can we make a non zero idempotent element of $R$ by this relation?

**3**

votes

**0**answers

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 $S(M)$.)

**2**

votes

**1**answer

183 views

### Deciding whether a non-f.g. non-divisible flat module is projective or not

Assume $S= R[T]/(f)= R[w]$ is a flat non-divisible $R$-module, where $R$ is a noetherian UFD, $T$ is an indeterminate over $R$, and $f\in R[T]$ is a non-monic polynomial of positive degree.
Can we ...

**0**

votes

**0**answers

36 views

### Deciding whether linear equations are solvable over specific subrings of $K(x_1,..,x_n)$

The definition of 'linear equations are solvable' which is meant here is
Let $R$ be a commutative ring (associative and with unity).
For given $m \in \mathbb{N}$ and $b \in R$,
it is decideable ...

**3**

votes

**2**answers

556 views

### Can the projective line be provided with a ring structure?

A definition of multiplication on the projective $1$-points $(a:b)$ of $P_K^1$ with $a$ and $b$ elements of a field $K$ ( e.g. the real or rational numbers ) can be given by mimicking the ...

**0**

votes

**1**answer

68 views

### Some references for f-ring

A commutative ring $R$ is said to be an $f-ring$ if every pure ideal is generated by idempotents. (Recall that the ideal $I$ is said to be pure if for each $a\in I$ there
is a $b\in I$ such that $ab = ...

**0**

votes

**3**answers

260 views

### Example of indecomposable self injective ring

Is there any example of an indecomposable self-injective commutative ring with 4 or more maximal ideals?$$$$$$$$

**3**

votes

**1**answer

153 views

### Example of a homogeneous (not monomial) $(x,y)$-primary ideal $I$ in $K[x,y]$

Is there any example of a homogeneous (not monomial) $(x,y)$-primary ideal $I$ in $K[x,y]$ such that $I$ is complete and and there exists a minimal reduction $J$ of $I$ such that $J\overline{I^n}=\...

**10**

votes

**1**answer

293 views

### Non-field example of a commutative, local, dual ring with nilradical $N$ such that $ann(N)\nsubseteq N$

I asked this question on math.stackexchange a month ago with no progress, even after a bounty. I hope to eliminate one if the other receives a satisfactory answer.
For an ideal $I\lhd R$ in a ...

**0**

votes

**0**answers

114 views

### book for help on problems with noetherian rings

Can you please introduce to me a book which would help me to prove the two following problems?
In a noetherian ring, every integrally closed ideal is unmixed.
Let $R$ be a noetherian ring, $P$ a ...

**7**

votes

**0**answers

338 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 Z\...

**1**

vote

**0**answers

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 $M_{\Delta}=\...

**0**

votes

**1**answer

144 views

### Change of grading used in the paper “The diagonal subring and the Cohen-Macaulay property of a multigraded ring” by Eero Hyry

I am currently reading the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring" by Eero Hyry. I do not understand the following part in Lemma 1.1. here.
Let $T=\...

**22**

votes

**2**answers

536 views

### What are applications of commutativity theorems for rings?

Herstein's little book "Noncommutative Rings" has a chapter called Commutativity Theorems in which he proves results like Jacobson's theorem: if a ring (associative with identity, please) has the ...

**0**

votes

**0**answers

81 views

### Idempotent ideal in ring of continuous functions

Is there any equivalence conditions under which an ideal $I$ in ring of continuous functions be be an idempotent ideal?

**3**

votes

**1**answer

256 views

### Castelnuovo-Mumford regularity in multigraded case

Let $R=\oplus_{n\geq 0}R_n$ be a standard Noetherian commuative graded ring over a local ring $(A,m)$ where $R_0=A.$ Put $R_+=\oplus_{n\geq 1}R_n.$ Let $M$ be a finitely generated $\mathbb Z$-graded $...

**3**

votes

**1**answer

137 views

### Can height one maximal ideals in the normalization contract to non-height one primes in the base?

Let $R$ be a local (Noetherian) integral domain of dimension greater than one. Can the integral closure (i.e. normalization) of $R$ have a maximal ideal of height one?

**3**

votes

**1**answer

128 views

### I need to refind a reference on multigraded Hilbert series

I found a theorem about multigraded Hilbert series stated as follows:
Let $R$ be a Noetherian multigraded algebra $R:=\bigoplus_{j\in\mathbb{N}^m}{R_j}$ over $R_0=\mathbb{C}$. If $R$ is generated by $...

**12**

votes

**3**answers

888 views

### The number of ideals in a ring

Here is a question that I first asked in math.stackexchange, but I think the question must be proposed here.
Let $R$ be a finite commutative ring with identity. Under what conditions the number ...

**2**

votes

**3**answers

314 views

### ideals of polynomial ring with complex number coefficients

Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$.
Let $f,g\in \mathbb{C}[x,y]$.
Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$.
...

**1**

vote

**1**answer

370 views

### ideals of polynomial ring of two variables generated by two elements

Let $f,g$ be two polynomials in $\mathbb{Z}[x,y]$, given by
$$
f(x,y)=x^4-3xy+y^2,$$
$$
g(x,y)=x^5-4xy+3xy^2.$$
Let $I=(f,g)$ be the ideal in $\mathbb{Z}[x,y]$ generated by $f$ and $g$.
Is $x,x^2,...

**1**

vote

**2**answers

394 views

### Surjectivity of trace map

Let $R$ be a closed integral domain with its fraction field $F$. Let $K$ be a finite separable extension field of $F$, and let $A$ be the integral closure of $R$ in $K$.
It is well known that the ...

**2**

votes

**2**answers

439 views

### Ideals generated by two elements in the polynomial ring of two variables over a field

Let $k$ be a field. For example, $k=\mathbb{Q}$ or $\mathbb{Z}/p$, $p$ prime.
Let $k[x,y]$ be the polynomial ring.
Let $f,g\in k[x,y]$.
Let the ideal $I=(f,g)$ be the ideal of $k[x,y]$ ...

**3**

votes

**2**answers

335 views

### Can there be a non-trivial epimorphism (of rings) from a field? [closed]

I apologize if this question is trivial, but I just cant figure it out. Let $K$ be a field and let $K\longrightarrow A$ be an epimorphism of rings. Is it necessary that $A=K$?

**1**

vote

**1**answer

201 views

### Necessary and sufficient condition for $can : A^X\otimes_A A^Y\rightarrow A^{X\times Y}$ to be an embedding

The two sets are, of course, supposed infinite.
This question is related to that one
Commutation of tensor products with inverse limits in a specific case
where it received a (partial) answer ($A$ ...

**1**

vote

**1**answer

144 views

### Structure of $\text{Aut}_R(R[X])$

Let $R$ be a commutative ring with identity. I'd like to know how to determine the set $\text{Aut}_R(R[X])$ of all $R$-automorphisms of $R[X]$.
I've proved that all $\sigma\in\text{Aut}_R(R[X])$ ...

**2**

votes

**0**answers

81 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 $\hat{A}=\varprojlim_{r,l}k[x_{1},x_{2},..]/(x_{1}^{r},..x_{l}^{r},x_{l+...

**0**

votes

**1**answer

252 views

### Commutation of tensor products with inverse limits in a specific case

For $X,Y$ sets, let's denote $Y^X$ the set of all mappings $X\rightarrow Y$. If $Y(=R)$ is a ring, $R^X$ is a $R$-module (well, a bi-module but my question is - at first - concerning commutative rings)...

**3**

votes

**0**answers

198 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 ...

**5**

votes

**1**answer

395 views

### When is $1+a+a^2+\dotsb+a^{{\rm ord}_n(a)-1}$ divisible by $n$?

I posted this question on math.SE 10 days ago and had no answer, comment or vote. If an answer is not available, I could really use a reference point as well.
For the sake of completeness, I am ...

**1**

vote

**0**answers

168 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 $\...

**6**

votes

**0**answers

148 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, ...

**1**

vote

**2**answers

173 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 ...

**9**

votes

**0**answers

153 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 ...

**3**

votes

**3**answers

239 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 ...

**1**

vote

**1**answer

139 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 ...

**0**

votes

**1**answer

132 views

### Canonical module of rees algebra

[Example 4.27, Integral Closure, Rees Algebras, Multiplicities, Algorithms] by Vasconcelos, says that if $I=(f_1,\ldots,f_g)$ is an ideal generated by a regular sequence with $g\ge 2$ then the ...

**2**

votes

**1**answer

110 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

**1**answer

221 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 ...

**0**

votes

**1**answer

144 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

**1**answer

300 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

**0**answers

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 $I$⊂$P$?

**5**

votes

**2**answers

258 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

**1**answer

162 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

**0**answers

84 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

**0**answers

74 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 ...