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Paper   IPM / M / 9537
School of Mathematics
  Title:   On prime modules and dense submodules
  Author(s):  M. Behboodi
  Status:   Published
  Journal: J. Commut. Algebra
  Vol.:  4
  Year:  2012
  Pages:   479-488
  Supported by:  IPM
Let R be a commutative ring with identity and let M be a unital R-module. A submodule N of M is called a dense submodule, if M=∑φφ(N) where φ runs over all the R-morphisms from N into M. An R-module M is called a π-module if every non-zero submodule is dense in M. This paper makes some observations concerning prime modules and π-modules over a commutative ring. It is shown that an R-module M is a prime module if and only if every nonzero cyclic submodule of M is a dense submodule of M. Moreover, for modules with nonzero socles and co-semisimple modules over any ring and for all finitely generated modules over a principal ideal domain (PID), the two concepts π and prime are equivalent. Rings R over which the two concepts π and prime are equivalent for all R-modules are characterized. Also, it is shown that if M is a π-module over a domain R with dim(R)=1, then either M is a homogeneous semisimple module or a torsion free module. In particular, if M is a multiplication module over a domain R with dim(R)=1, then M is a π-module if and only if either M is a simple module or R is a Dedekind domain and M is a faithful R-module.

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