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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 $\mathcal{P}... 0answers 84 views ###$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.,$a_n\cdots a_1=0$).... 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. ... 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 ... 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 ... 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$...
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 \...