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Kernels of Morphisms

In: Semirings and their Applications

Author

Listed:
  • Jonathan S. Golan

    (University of Haifa)

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.

Suggested Citation

  • Jonathan S. Golan, 1999. "Kernels of Morphisms," Springer Books, in: Semirings and their Applications, chapter 10, pages 121-127, Springer.
    • Handle: RePEc:spr:sprchp:978-94-015-9333-5_10
    DOI: 10.1007/978-94-015-9333-5_10
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