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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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