# Tagged Questions

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### 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 ...
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### Example of indecomposable self injective ring

Is there any example of an indecomposable self-injective commutative ring with 4 or more maximal ideals?
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### 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 ...
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### 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?
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### 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 ...
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### 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 $f,g$. ...
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### 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)...
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### 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 ...
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### 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 ...
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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 $\... 0answers 148 views ### flatness and derived completion Let$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$is not noetherian, ... 2answers 173 views ### if$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$be Gorenstein or Complete ... 0answers 153 views ### Weierstrass division theorem for henselian rings Let$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$is analytically normal and of ... 3answers 239 views ### canonical module can be identified with an ideal. how can one reach that ideal? Let$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$. So we ... 1answer 139 views ### Graded version of Baer's Criterion Baer's Criterion for injectiveness of modules says: "An$R$-module$E$is injective iff for all ideals$I$of$R$, every homomorphism$f\colon I \to E$can be extended to$R$." I wonder if there is a ... 1answer 132 views ### Canonical module of rees algebra [Example 4.27, Integral Closure, Rees Algebras, Multiplicities, Algorithms] by Vasconcelos, says that if$I=(f_1,\ldots,f_g)$is an ideal generated by a regular sequence with$g\ge 2$then the ... 1answer 110 views ### Matching power series to infinity As pointed out by Makoto, on this question about power series rings and the axiom of choice, an idea I had needed the axiom of dependent choice to work. However, the construction raises another ... 1answer 221 views ### finiteness dimension$R$is a local Noetherian ring.$f_I(M)$, the finiteness dimension of a module$M$relative to$I$, is defined in ... 1answer 144 views ### if$ \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$, can one ... 1answer 300 views ### intuitive interpretation of analytic spread I am studying analytic spreads from Bruns-Herzog's book. The definition is clear but calculation of the analytic spread of an ideal is hard for me in practice. I wonder if it is hard for you too. ... 0answers 144 views ### On Prüfer domains Is there any Prüfer domain$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$? 2answers 258 views ### Integral domains with totally ordered spectra In my research I ended up trying to prove some properties of integral domains such that their spectrum is a totally ordered poset. Are there some nice (ubiqitous/natural) examples of such domains, ... 1answer 162 views ### Bounded Index of Nilpotency of$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$be a commutatitve ring with bounded index$...
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 ...