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On a Class of Perfect Rings

Published online by Cambridge University Press:  20 November 2018

Vlastimil Dlab
Vlastimil Dlab*
Affiliation:
Carleton University, Ottawa, Ontario
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In [3], the perfect rings of Bass [1] were characterized in terms of torsions in the following way:

A ring R is right perfect if and only if every (hereditary) torsion in the categoryMod Rof all left R-modules is fundamental (i.e. generated by some minimal torsions) and closed under taking direct products; as a consequence, the number of all torsions inMod Ris finite and equal to 2n for a natural n.

Here, we present a simple description of those rings R which allow only two (trivial) torsions, viz. 0 and Mod R (and thus, are right perfect by [3]). Finite direct sums of these rings represent a natural generalization of completely reducible (i.e. artinian semisimple) rings (cf. Theorem 2) and we shall call them for that matter π-reducible rings. They can also be characterized in terms of their idempotent two-sided ideals.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 4 , 01 August 1970 , pp. 822 - 826
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Bass, H., Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488.Google Scholar
2. Courter, R., Finite direct sums of complete matrix rings over perfect completely primary rings, Can. J. Math. 21 (1969), 430–446.Google Scholar
3. Dlab, V., A characterization of perfect rings, Pacific J. Math. 83 (1970), 79–88.Google Scholar
4. Findlay, G. D. and Lambek, J., A generalized ring of quotients. I, II, Can. Math. Bull. 1 (1958), 77-85, 155–167.Google Scholar