Extensions of semimodules and the Takahashi functor $Ext_{\Lambda}(C, A)$

Let $\Lambda$ be a semiring with 1. By a Takahashi extension of a $\Lambda$-semimodule $X$ by a $\Lambda$-semimodule $Y$ we mean an extension of $X$ by $Y$ in the sense of M. Takahashi [10]. Let $A$ be an arbitrary $\Lambda$-semimodule and $C$ a $\Lambda$-semimodule which is normal in Takahashi's sense, that is, there exist a projective $\Lambda$-semimodule $P$ and a surjective $\Lambda$-homomorphism $\varepsilon : P \to C$ such that $\varepsilon$ is a cokernel of the inclusion $\mu:\operatorname{Ker}(\varepsilon)\hookrightarrow P$. In [11], following the construction of the usual satellite functors, M. Takahashi defined $\operatorname{Ext}_{\Lambda}(C,A)$ by\[ \operatorname{Ext}_{\Lambda}(C,A)=\operatorname{Coker}(\operatorname{Hom}_{\Lambda}(\mu,A)) \]and used it to characterize Takahashi extensions of normal $\Lambda$-semimodules by $\Lambda$-modules.

In this paper we relate $\operatorname{Ext}_{\Lambda}(C,A)$ with other known satellite functors of the functor $\operatorname{Hom}_{\Lambda}(-,A)$.

Keywords

semiring, semimodule, projective semimodule, normal semimodule, extension of semimodule, satellite functor

2010 Mathematics Subject Classification

16Y60, 18E25, 18Gxx, 20M50

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Published 1 January 2003