  We show that any $n$-absorbing ideal must be strongly $n$-absorbing, which is
the first of Anderson and Badawi's three interconnected conjectures on
absorbing ideals. We prove this by introducing and studying objects called
maximal and semimaximal collections, which are tools for analysing the
multiplicative ideal structure of commutative rings. We also make heavy use of
previous work on absorbing ideals done by Anderson and Badawi, Choi and Walker,
Laradji, and Donadze, among others. At the end of this paper we pose three
conjectures, each of which imply the last unsolved conjecture made by Anderson
and Badawi.
