Additive Ternary $\Gamma$-Modules and Homological Algebra

Fix a commutative monoid $(T,+,0)$, a commutative monoid $(\Gamma,+,0_\Gamma)$, and a map $$(a,\alpha,b,\beta,c)\longmapsto a\,\alpha\,b\,\beta\,c\in T$$ which is additive in each variable and associative in the ternary sense. A left additive ternary $\Gamma$-module is an abelian group $M$ equipped with an action $(a,\alpha,b,\beta,m)\mapsto a\,\alpha\,b\,\beta\,m$ satisfying the same associativity constraints. Two scalars are intrinsic, so the action is genuinely polyadic; in particular, we do not assume a canonical unary action $T\times M\to M$. The first part constructs the \emph{operator ring} $\mathcal O_{T,\Gamma}$ generated by the left translations $(a,\alpha,b,\beta)$. It is shown that $T\!\!\;\!-\!\Gamma\mathrm{Mod}$ is equivalent to the ordinary module category $\mathcal O_{T,\Gamma}\mathrm{Mod}$. The category is therefore abelian. Under the unital operator hypothesis \emph{(U)} (i.e. when $\mathcal O_{T,\Gamma}$ is unital), it has enough projectives and enough injectives. The second part defines a tensor product over $T$ by a \emph{bi-balanced} universal property forced by the ternary action, proves right exactness of $-\otimes_T-$, and establishes a Tensor--Hom adjunction for bimodules. Assuming \emph{(U)}, derived functors $\mathrm{Ext}$ and $\mathrm{Tor}$ are developed inside the additive track. A finite example with $T=\ZZ/4\ZZ$ gives explicit computations $$\mathrm{Ext}^1_T(\ZZ/2\ZZ,\ZZ/2\ZZ)\cong \ZZ/2\ZZ,\qquad \mathrm{Tor}^T_1(\ZZ/2\ZZ,\ZZ/2\ZZ)\cong \ZZ/2\ZZ,$$ with a concrete nonsplit extension and an explicit failure of tensor exactness. Counterexamples isolate the precise points where naive binary-module arguments break.