Wat exactly does the notation $A : (BC)$ mean? We are talking about fractional R ideals. From definitions i get, if $a \in A:(BC)$ then $a(BC)\subset A$
But what am i saying exactly?
I hope someone can explain this to me.
Wat exactly does the notation $A : (BC)$ mean? We are talking about fractional R ideals. From definitions i get, if $a \in A:(BC)$ then $a(BC)\subset A$
But what am i saying exactly?
I hope someone can explain this to me.
The usual framework is: $\,R\,$ is an integer domain, $\,K\,$ is its fractions fields, $\,A,B,C\,$ are fractional ideals, i.e.: $\,A\,$ is a fractional ideal if it is a $\,K-\,$ module for which there exists $\,0\neq r\in R\,$ s.t. $\,rA\subset R\,$ .
So by $\,A:BC\,$ you most probably meaning the set of all elements $\,t\in K\,$ s.t. $\,t(BC)\subset A\,$ .
The following is borrowed from Atiyah-Macdonald:
If $N,P$ are submodules of $M$, we define $(N:P)$ to be the set of all $a\in A$ such that $aP\subset N$. It is an ideal of $A$. In particular, $(0:M)$ is the set of all $a\in A$ such that $aM=0$.