In this article, the notion of purely large structure is introduced. It is shown, with the aid of a Theorem of Rothmaler, that any
finitely accessible class possesses purely large structures. This applies to the class Mod(R) of all left modules over a given
ring R. The theory T∗ of purely large modules is always complete. It is shown that T∗ is model-complete if and only if R
is regular. For any algebra of finite representation type R, over an infinite field, T∗ is axiomatizable by one sentence over
Th(Mod(R)). A characterization of pure semisimple rings, in terms of purely large modules, is obtained. |
