The tag has no usage guidance.

learn more… | top users | synonyms

12
votes
1answer
527 views

Duality between topology and bornology

I want to understand in what sense topology is dual to bornology at a most basic level. Therefore, I rephrased the definition of a bornology in the following way: Let $X$ be a set and let $\mathcal{P}...
2
votes
0answers
84 views

$T$-nilpotent ideals

Recall that a subset $I$ of a ring $R$ is left (resp., right) $T$-nilpotent in case for every sequence $$a_1,a_2,\cdots $$ in $I$ there is an $n$ such that $a_1\cdots a_n=0$ (resp., $a_n\cdots a_1=0$)....
2
votes
0answers
60 views

some sort of 'saturation' of module quotients

Let $R$ be a local Noetherian ring over a field, with the maximal ideal $\mathfrak{m}$. (e.g. $R=k[[x_1,\dots,x_{p>1}]]$) Given two $R$-modules, $N\subset M$, of the same (finite, non-zero) rank. ...
1
vote
0answers
46 views
+50

How are Polynomials of Toric ideals Studied with Exponents as ST-cuts?

Topic: Toric ideals on Expected value of Structure Functions in Random Graphs Goal: to understand the toric ideal where the exponents $h_i$ and $s_j$ are st-vertex-cuts of a digraph \begin{equation} ...
0
votes
1answer
147 views

In what conditions every ideal is an extension ideal? Is every prime ideal extension of prime ideal?

Let $R$ and $S$ be commutative rings (with $1$), and $f : R\to S$ be a ring homomorphism. For an ideal $I$ of $R$, set $I^e:=\langle f(I)S\rangle$ (called the extension of $I$ to $S$). When $f$ is ...
0
votes
0answers
101 views

Can it occur that $q^{ce}$ is a prime ideal (of $S$), while $q^{ce}\neq q $?

Let $R$ and $S$ be commutative rings (with $1$) and $f : R\to S$ be a ring homomorphism. For an ideal $I$ of $R$, set $I^e:=\langle f(I)S\rangle$ (called the extension of $I$ to $S$) and for an ideal $...
-1
votes
0answers
38 views

The definition of primary ideal [closed]

The definition from the book is: "An ideal $I$ of a commutative ring $A$ is primary if and only if $I\neq A$ and $\forall x,y\in A$ with $x\cdot y\in I$ and $x\notin I $$\Rightarrow \exists m\in \...