Vol. 27, No. 3, 1968

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Quasi-isomorphism and TFM rings

William Jennings Wickless

Vol. 27 (1968), No. 3, 633–639

Two rings A and B are quasi-isomorphic if and only if there exist ideals Aand Bcontained in A and B respectively such that ABas rings and A∕Aand B∕Bare 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.

Mathematical Subject Classification
Primary: 16.40
Received: 4 January 1968
Published: 1 December 1968
William Jennings Wickless