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Extensions of semi-hereditary rings

Published online by Cambridge University Press:  09 April 2009

M. W. Evans
M. W. Evans
Affiliation:
84 Glencairn Ave., East Brighton, 3187, Australia.
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Abstract

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Hattori (1960) defined a right R-module A to be torsion-free if for all a ∈ A and x ∈ R, ax = 0 implies that there exist elements {x1, x2, …, xn} ⊆ R with xix = 0 for all 1 ≦ i ≦ n and {a1, a2, …, an} ⊆ A such that a = aixi. Left torsion-free is defined similarly. It is shown that for a ring R, these torsion-free modules are the torsion-free class of a hereditary torsion theory, corresponding to a perfect topology, if and only if the left flat epimorphic hull of R is a regular ring which is both left and right torsion-free. A class of right semi-hereditary rings for which the torsion-free modules of Hattori satisfy the above property are found and this class of rings is discussed.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 23 , Issue 3 , May 1977 , pp. 333 - 339
Copyright
Copyright © Australian Mathematical Society 1977

References

Bass, Hyman (1960), ‘Finistic dimension and a homological generalization of semi-primary rings’, Trans. Amer. Math. Soc. 95, 466–488.CrossRefGoogle Scholar
Cartan, Henri and Eilenberg, Samuel (1956), Homological Algebra (Princeton University Press, Princeton, New Jersey, 1956).Google Scholar
Endo, Shizuo (1960), ‘Note on p.p. rings (A supplement to Hattori's paper)’, Nagoya Math. J. 17, 167–170.CrossRefGoogle Scholar
Endo, Shizuo (1961), ‘On semi-hereditary rings’, J. Math. Soc. Japan 13, 109–119.CrossRefGoogle Scholar
Evans, M. W. (1972), ‘On commutative P.P. rings’, Pacific J. Math. 41, 687–697.CrossRefGoogle Scholar
Findlay, George D. (1970), ‘Flat epimorphic extensions of rings’, Math. Z. 118, 281–288.CrossRefGoogle Scholar
Goldman, Oscar (1969), ‘Rings and Modules of quotients’, J. Algebra 13, 10–47.CrossRefGoogle Scholar
Akira, Hattori (1960), ‘A foundation of torsion theory for modules over general rings’, Nagoya Math. J. 17, 147–158.Google Scholar
Hinohara, Yukitoshi (1960), ‘Note on non-commutative semi-local rings’, Nagoya Math. J. 17, 161–166.CrossRefGoogle Scholar
Jøndrup, S. (1970), ‘On finitely generated flat modules II’, Math. Scand. 27, 105–112.CrossRefGoogle Scholar
Jøndrup, Søren (1971), ‘p.p. rings and finitely generated flat ideals’, Proc Amer. Math. Soc. 28, 431–435.Google Scholar
Knight, J. T. (1970), ‘On epimorphisms of non-commutative rings’, Proc. Cambridge Philos. Soc. 68, 589–600.CrossRefGoogle Scholar
Lambek, Joachim (1966), Lectures on Rings and Modules (Blaidsell, Waltham, Massachusetts; London; Toronto; 1966).Google Scholar
Quentel, Y. (1971), ‘Sur la compactie du spectre minimal d'un anneau’, Bull. Soc. Math. France 99, 265–272.CrossRefGoogle Scholar
Sandomierski, Francis L. (1968), ‘Nonsingular rings’, Proc. Amer. Math. Soc. 19, 225–230.CrossRefGoogle Scholar
Small, L. (1966), ‘Hereditary rings’, Proc. Nat. Acad. Sci. USA 55, 25–27.CrossRefGoogle ScholarPubMed
Stenström, Bo (1970), ‘On the completion of modules in an additive topology’, J. Algebra 16, 523–540.CrossRefGoogle Scholar
Stenström, Bo (1971), Rings and Modules of Quotients (Lecture Notes in Mathematics, 237. Springer-Verlag, Berlin, Heidelberg, New York, 1971).CrossRefGoogle Scholar
Storrer, Hans H. (1971), ‘Rings of quotients of perfect rings’, Math. Z. 122, 151–165.CrossRefGoogle Scholar
Wiegand, Roger (1971), ‘Modules over universal regular rings’, Pacific J. Math. 39, 807–819.CrossRefGoogle Scholar