Factor Semimodules

Jonathan S. Golan
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Jonathan S. Golan: University of Haifa

Chapter 15 in Semirings and their Applications, 1999, pp 163-179 from Springer

Abstract: Abstract Congruence relations played an important role in the theory of semirings and we would expect them to play a similar role in the theory of semimodules. Let R be a semiring and let M be a left R-semimodule. An equivalence relation ρ on M is an R-congruence relation if and only if m ρ m′ and n ρ n′ in M imply that (m + n) ρ (m′ + n′) and rm ρ rm′ for all r ∈ R. In other words, an R-congruence relation ρ on M is an equivalence relation satisfying the condition that ρ is also a subsemimodule of M × M. Denote the set of all incongruence relations on M by R — cong(M). This set is nonempty since it contains the trivial R -congruence ≡t defined by m ≡t m′ if and only if m = m′ and the universal R -congruence ≡u defined by m ≡u m′ for all m, m′ ∈ M. If M ≠ {0 m } and these are the only two elements of R — cong(M), then the R-semimodule M is simple. Moreover, R — cong(M) is partially-ordered by the relation ≤ defined by ρ ≤ ρ′ if and only if m ρ m′ implies that m ρ′ m′. Clearly ≡t ≤ ρ ≤ ≡u for all incongruence relations ρ in R — cong(M).

Keywords: Equivalence Relation; Scalar Multiplication; Complete Lattice; Primitive Element; Congruence Relation (search for similar items in EconPapers)
Date: 1999
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DOI: 10.1007/978-94-015-9333-5_15

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