“School of Mathematics”
Back to Papers HomeBack to Papers of School of Mathematics
Paper IPM / M / 763  


Abstract:  
A Euclidian is a natural extension of a Euclidian ring. Any
nonzero submodule of a Euclidian Rmodule is a cyclic
Rmodule. It is shown that a torsion free cyclic Rmodule 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 Rmodule has no universal side divisors if
and only if R has no universal divisors. Also, a torsion free
cycbic Rmodule with no universal side divisors over an integral
domain can never be a Euclidian Rmodule. Stable Rmodules are
defined and it is shown that any torsion free cyclic Rmodule is
stable if and only if R is a stable ring. Finally, it is shown
that a stable torsion free cyclic Rmodule over a principal
ideal domain is a Euclidian Rmodule.
All rings (unless otherwise indicated) are commutative rings with
identity and modules are unitary modules.
Download TeX format 

back to top 