Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-08T05:11:12.952Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1]Bland, P.E., “Perfect torsion theories”, Proc. Amer. Math. Soc. 41 (1973), 349–355.CrossRefGoogle Scholar
[2]Dickson, Spencer E., “A torsion theory for abelian categories”, Trans. Amer. Math. Soc. 121 (1966), 223–235.CrossRefGoogle Scholar
[3]Goldman, Oscar, “Rings and modules of quotients”, J. Algebra 13 (1969), 10–47.CrossRefGoogle Scholar
[4]Jans, J.P., “Some aspects of torsion”, Pacific J. Math. 15 (1965), 1249–1259.Google Scholar
[5]Lambek, Joachim, Torsion theories, additive semantics and rings of quotients (Lecture Notes in Mathematics, 177. Springer-Verlag, Berlin, Heidelberg, New York, 1971).Google Scholar
[6]Rangaswamy, K.M., “Codivisible modules”, Comm. Algebra 2 (1974), 475–489.CrossRefGoogle Scholar
[7]Stenström, Bo, Rings and modules of quotients (Lecture Notes in Mathematics, 237. Springer-Verlag, Berlin, Heidelberg, New York, 1971).CrossRefGoogle Scholar