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Paper   IPM / M / 763
School of Mathematics
  Title:   Euclidian modules
  Author(s):  A. M. Rahimi
  Status:   To Appear
  Journal: Libertas Math.
  Supported by:  IPM
  Abstract:
A Euclidian is a natural extension of a Euclidian ring. Any nonzero submodule of a Euclidian R-module is a cyclic R-module. It is shown that a torsion free cyclic R-module is Euclidian if and only if R is a Euclidian ring. The concept of side divisors and universal side divisors in a ring are generalized and studied in the cyclic modules. It is shown that a torsion free cyclic R-module has no universal side divisors if and only if R has no universal divisors. Also, a torsion free cycbic R-module with no universal side divisors over an integral domain can never be a Euclidian R-module. Stable R-modules are defined and it is shown that any torsion free cyclic R-module is stable if and only if R is a stable ring. Finally, it is shown that a stable torsion free cyclic R-module over a principal ideal domain is a Euclidian R-module.
All rings (unless otherwise indicated) are commutative rings with identity and modules are unitary modules.

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