Kernels of Morphisms

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

Chapter 10 in Semirings and their Applications, 1999, pp 121-127 from Springer

Abstract: Abstract By Proposition 9.8 we see that if γ: R → S is a morphism of semirings then γ-1(0) is an ideal of R, called the kernel of γ, and denoted by ker(γ). By Proposition 9.46, ker(γ) is an ideal of R. If R is a ring, we know that any ideal of R can be the kernel of a morphism from R to some ring S but, as we shall see, this is not the case for arbitrary semirings. Also, unlike the case of rings, we note that a morphism of semirings γ: R → S need not be monic when ker(γ) = {0}. To see an example of this, consider the totally-ordered set R = {0, a, 1} on which we define addition to be max and multiplication to be min. This is a semiring by Example 1.5. Let γ: R → B be the character of R denned by γ(0) = 0 and γ(a) = γ(1) = 1. This map has kernel {0} but is not monic.

Keywords: Normal Subgroup; Left Ideal; Subdirect Product; Surjective Morphism; Normal Divisor (search for similar items in EconPapers)
Date: 1999
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DOI: 10.1007/978-94-015-9333-5_10

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