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

Abstract

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 $\varepsilon : P \longrightarrow C$ such that $\varepsilon$ is a cokernel of the inclusion $\mu:\rm{Ker}(\varepsilon)\hookrightarrow P$. In [11], following the construction of the usual satellite functors, M. Takahashi defined $\rm{Ext}_{{}_\Lambda}(C,A)$ by

$$ \rm{Ext}_{{}_\Lambda}(C,A)=\rm{Coker}(\rm{Hom}_{{}_\Lambda}(\mu,A))$$

and used it to characterize Takahashi extensions of normal $\Lambda$-semimodules by $\Lambda$-modules.

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