Strongly Rigid and I-Rigid Rings

In this paper we consider questions connected with the problem of rigidity of rings for the class of finite-rank torsion-free rings (A ring is called rigid, if it has only trivial endomorphisms). We study strongly rigid and I-rigid rings [3] (A ring R is called strongly rigid, if any ring quasi-isomorphic to R is rigid). The main results of this paper are the characterization of strongly rigid rings and the establishment of relations between strongly rigid and I-rigid rings. It is shown that the verification of strong rigidity of the ring reduces to testing the rigidity of some field of algebraic numbers over its subfield. This gives rise to examples of non-trivial strongly rigid rings.