# Extensions of Semimodules and The Takahashi Functor ExtΛ(C,A)

## Alex Patchkoria

Let $\Lb$ be a semiring with 1. By a Takahashi extension of a $\Lb${\mbox-se}\-mi\-module $X$ by a $\Lb$-semimodule $Y$ we mean an extension of $X$ by $Y$ in the sense of M. Takahashi [{\bf 10}]. Let $A$ be an arbitrary $\Lb${\mbox -se}\-mi\-module and $C$ a $\Lb$-semimodule which is normal in Takahashi's sense, that is, there exist a projective $\Lb$-semimodule $P$ and a surjective $\Lb$-homomorphism $\xymatrix{\ve:P\ar[r]& C}$ such that $\ve$ is a cokernel of the inclusion $\mu:\Ker(\ve)\hookrightarrow P$. In [{\bf 11}], following the construction of the usual satellite functors, M. Takahashi defined $\Ext_{{}_\Lb}(C,A)$ by

$$\Ext_{{}_\Lb}(C,A)=\Coker(\Hom_{{}_\Lb}(\mu,A))$$

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

In this paper we relate $\Ext_{{}_\Lb}(C,A)$ with other known satellite functors of the functor $\Hom_{{}_\Lb}(-,A)$.

Homology, Homotopy and Applications, Vol. 5(2003), No. 1, pp. 387-406

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