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Paper   IPM / M / 8282
School of Mathematics
  Title:   Some Commutative ring results extended to unitary semimodules over commutative semirings
  Author(s):  A. M . Rahimi
  Status:   To Appear
  Journal: Libertas Math.
  Supported by:  IPM
All semirings are commutative semirings with identity elements 0 and 1. Let A be a unitary semimodule on a commutative semirings R. Some basic algebraic properties of subsemimodules are investigated. It is shown that a semimodule A satisfies the ascending chain condition on subsemimodules if and only if every subsemimodule of A is finitely generated. The intersection of all maximal subsemimodules of a semimodule A is defined to be the Jaconson radical of A. It is shown that every proper subsemimodule of a finitely generated semimodule A is contained in a maximal subsemimodule of A. Minimal generating sets, rank and the stable range of semimodules are defined. In a semimodule A, every element outside the Jaconson radical of A belongs to a minimal generating set of A whenever A satisfies the ascending chain condition on subsemimodules. It is shown that for any positive integer n and any n-stable semimodule A, rank(A) < n. A version of Nakayama's lemma for finitely generated semimodules is proved. Prime, primary, and the radical of subsemimodules are defined and some of their properties by applying the injector ideals of R, a Chinese remainder theorem is proved for R-semimodules.

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