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On flat finitely generated ideals

Published online by Cambridge University Press:  17 April 2009

David E. Dobbs
David E. Dobbs
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37916, USA.
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Abstract

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It is shown that if I is a finitely generated ideal of a commutative ring R such that the multiplication map I ⊗RI → I is an injection, then I is locally principal. As a corollary, one obtains a new homological characterization of Prüfer domains.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 21 , Issue 1 , February 1980 , pp. 131 - 135
Copyright
Copyright © Australian Mathematical Society 1980

References

[1]Bourbaki, Nicolas, Elements of mathematics. Commutative algebra (Hermann, Paris; Addison-Wesley, Reading, Massachusetts, 1972).Google Scholar
[2]Chase, Stephen U., “Direct products of modules”, Trans. Amer. Math. Soc. 97 (1960), 457–473.CrossRefGoogle Scholar
[3]Davis, Edward D., “A remark on Priifer rings”, Proc. Amer. Math. Soc. 20 (1969), 235–237.Google Scholar
[4]Endo, Shizuo, “On flat modules over commutative rings”, J. Math. Soc. Japan 14 (1962), 284–291.Google Scholar
[5]Gilmer, Robert W., Multiplicative ideal theory, Revised edition (Pure and Applied Mathematics, 12. Marcel Dekker, New York, 1972).Google Scholar
[6]Gilmer, Robert and Heinzer, William, “On the number of generators of an invertible ideal”, J. Algebra 14 (1970), 139–151.CrossRefGoogle Scholar
[7]Heitmann, Raymond C., “Generating ideals in Prüfer domains”, Pacific J. Math. 62 (1976), 117–126.CrossRefGoogle Scholar
[8]Lambek, J., “A module is flat if and only if its character module is injective”, Canad. Math. Bull. 7 (1964), 237–243.CrossRefGoogle Scholar
[9]Richman, Fred, “Generalized quotient rings”, Proc. Amer. Math. Soc. 16 (1965), 794–799.CrossRefGoogle Scholar
[10]Sally, Judith D. and Vasconcelos, Wolmer V., “Stable rings”, J. Pure Appl. Algebra 4 (1974), 319–336.CrossRefGoogle Scholar
[11]Schülting, Heinz-Werner, “Über die Erzeugendenanzahl invertierbarer Ideale in Prüferringen”, Comm. Algebra 7 (1979), 1331–1349.CrossRefGoogle Scholar
[12]Smith, William W., “Invertible ideals and overrings”, Houston J. Math. 5 (1979), 141–153.Google Scholar