Factor Semirings

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

Chapter 8 in Semirings and their Applications, 1999, pp 95-103 from Springer

Abstract: Abstract In the category of rings, factor objects are determined by ideals. In the category of semirings, as in the category of lattices, this is not so and we must look instead at congruence relations. An equivalence relation ρ defined on a semiring R which satisfies the additional condition that if r ρ r′ and s ρ s′ in R then r + s ρ r′ + s′ and rs ρ r′s′ is called a congruence relation. The congruence relation ρ defined by r ρ r′ if and only if r = r′ is the trivial congruence relation on R. All other congruence relations on R are nontrivial. The congruence relation ρ defined by r ρ r′ for all r,r′ ∈ R is the improper congruence relation on R. All other congruence relations are proper. Note that ρ is improper if and only if 1 ρ 0. Indeed, if ρ is improper this is clearly true. Conversely, if 1 ρ 0 then for each r ∈R we have r = r1 ρ r0 = 0 and so ρ is improper.

Keywords: Prime Ideal; Maximal Ideal; Congruence Relation; SEMIPRIME Ideal; Nonempty Family (search for similar items in EconPapers)
Date: 1999
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DOI: 10.1007/978-94-015-9333-5_8

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