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Free, Projective, and Injective Semimodules

In: Semirings and their Applications

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  • Jonathan S. Golan

    (University of Haifa)

Abstract

Let R be a semiring and let M be a left R-semimodule. If A is a nonempty subset of M then there exists an R-homomorphism α: R (A)→ M defined by Σm∈Af(m)m. The set A is a set of generators for M precisely when this R-homomorphism is surjective. Moreover, α induces an R-congruence relation = α on R (A) as defined in Example 15.1. The set A is linearly independent if and only if ≡ α is the trivial relation, i.e. if and only if Σm∈Af(m)m=Σm∈Ag(m)m implies that f = g. If A is not linearly independent then it is linearly dependent. A linearly-independent set of generators for M is a basis of M over R. We note that if A is linearly dependent and if B ⊂ A then the subsemimodules of M generated by B and A \ B have no nozero element in common.

Suggested Citation

  • Jonathan S. Golan, 1999. "Free, Projective, and Injective Semimodules," Springer Books, in: Semirings and their Applications, chapter 17, pages 191-202, Springer.
    • Handle: RePEc:spr:sprchp:978-94-015-9333-5_17
    DOI: 10.1007/978-94-015-9333-5_17
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