  We explore some concepts of module theory that derive from the notion of
primeness, such as first modules, and extend them to more general environments.
We also provide descriptions of simple right semiartinian rings, right local
rings, semisimple rings, and simple rings in terms of their $\mathscr{A}$-first
modules with respect to a preradical class. We give examples of lattices that
can act on or can be acted by other posets (partially ordered sets).

Rings and Algebras