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Relatively flat modules

Published online by Cambridge University Press:  17 April 2009

Paul E. Bland
Paul E. Bland
Affiliation:
Department of Mathematics, Eastern Kentucky University, Richmond, Kentucky, USA.
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Abstract

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If (A′, B′), (B′, C′) and (A, B), (B, C) are torsion-torsion free theories on RM and MR respectively which are generated by an idempotent ideal I of R, then M ∈ RM is said to be relatively flat if (·) ⊗ RM preserves short exact sequences 0 → L → X → N → 0 in MR with N ∈ B. Several characterizations of relatively flat modules are given and it is shown that any module M ∈ RM which is codivisible with respect to (A′, B′) is relatively flat. In addition, when (A′, B′) is hereditary, it is proven that M ∈ RM is relatively flat if and only if M/IM is a flat R/I-module. Finally, a relatively flat dimension for M ∈ RM and a left global relatively flat dimension for R are defined and it is shown, again when (A′, B′) is hereditary, that the left global relatively flat dimension of R coincides with the left global flat dimension of R/I.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 13 , Issue 3 , December 1975 , pp. 375 - 387
Copyright
Copyright © Australian Mathematical Society 1975

References

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