Two rings A and B are
quasiisomorphic if and only if there exist ideals A′ and B′ contained in A and B
respectively such that A′≅B′ as rings and A∕A′ and B∕B′ are of bounded order as
abelian groups. A ring is TFM if and only if it only admits torsion free abelian
groups as irreducible modules. It is shown that quasiisomorphic TFM rings have
exactly the same abelian groups as irreducible modules. Several examples of TFM
rings are given.
A classification of TFM rings is given. The following results are obtained:
(1) A is TFM if and only if A∕pA is radical for all primes p.
(2) A is TFM if and only if A∕N is torsion free and no maximal modular right
ideal is dense in the subgroup lattice of A∕N, where N is the Jacobson Radical of
A.
(3) If A∕N divisible then A is TFM. The converse holds under the assumption of
minimum condition.
(4) A∕D radical ⇒ A TFM ⇒ A∕D has no nonzero idempotents, where D is the
maximal divisible subgroup of A. These conditions are equivalent under the
assumption of minimum condition.
Finally, the questions of the existance of a TFM radical and the determination of
the unique maximal TFM ideal of a ring are discussed.
