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| Paper IPM / M / 8282 |
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| Abstract: | |
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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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