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Two rings A and B are
quasi-isomorphic 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 quasi-isomorphic 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.
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