On the order of a module

Document Type : Research Paper

Authors

1 Department of‎ ‎Mathematics‎, ‎Vali-e-Asr University of Rafsanjan‎, ‎P.O‎. ‎Box 7718897111‎, ‎Rafsanjan‎, ‎Iran.

2 Department of‎ ‎Mathematics‎, ‎Vali-e-Asr University of Rafsanjan‎, ‎P.O‎. ‎Box 7718897111‎, ‎Rafsanjan‎, ‎Iran.

Abstract

Abstract. Let $(R,P)$ be a Noetherian unique factorization domain (UFD) and M be a finitely generated R-module. Let I(M)
be the first nonzero Fitting ideal of M and the order of M, denoted $ord_R(M)$, be the largest integer n such that $I(M) ⊆ P^n$. In this paper, we show that if M is a module of order one, then either M is isomorphic with direct sum of a free module and a cyclic module or M is isomorphic with a special module represented in the text. We also assert some properties of M while $ord_R(M) = 2.$

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Main Subjects

References

N. Bourbaki, Commutative Algebra, Springer-Verlag, Berlin, 1998.
D. A. Buchsbaum and D. Eisenbud, What makes a complex exact? J. Algebra 25 (1973) 259--268.
D. Eisenbud, Commutative Algebra with a View toward Algebraic Geometry, Springer-verlag, New York, 1995.
S. Hadjirezaei and S. Hedayat, On the first nonzero Fitting ideal of a module over a UFD, Comm. Algebra 41 (2013), no. 1, 361--366.
V. Kodiyalam, Integrally closed modules over two-dimensional regular local ring, Trans. Amer. math. soc. 347 (1995), no. 9, 3551--3573.
J. Lipman, On the Jacobian ideal of the module of differentials, Proc. Amer. Math. Soc. 21 (1969) 423--426.
J. Ohm, On the first nonzero Fitting ideal of a module, J. Algebra 320 (2008), no. 1, 417--425.
Y. Tiras and M. Alkan, Prime modules and submodules, Comm. Algebra 31 (2003), no. 11, 5253--5261.

History

• Receive Date: 14 May 2014
• Revise Date: 31 May 2015
• Accept Date: 02 June 2015