Graph Module Theory in the Wang Framework — Endogenous Linear Algebra over the Graph Ring

Abstract This paper develops the second algebraic layer following the graph ring construction in the Wang framework. The base ring is the graph ring R_: addition is induced by disjoint union, multiplication is induced by the closed‑neighborhood tensor product, equivalently the strong product of finite simple graphs, and the ring is canonically a polynomial ring over Z on connected strong‑prime graph variables. A graph module is, in its ambient form, a module over this graph‑born ring. The endogenous content is not the assertion that ordinary module theory is new, but the insistence that the scalar ring, the preferred generators, and the representation maps are acquired from graph‑level data before external algebra is used. The paper distinguishes three layers. First, the ambient category Modₑ consists of all R-modules and is an abelian category. Secondly, graph‑presented modules are cokernels of maps between free modules generated by graph atoms: connected graph classes, support‑rooted observations, row‑orbit packets, scanning‑table atoms, weighted packets, or affine‑scheme coordinates from the earlier Wang papers. Thirdly, representation modules arise only after an audited ring homomorphism from R to a target algebra has been supplied. This separation prevents a common undefined‑action error: a complete invariant, a finite‑dimensional operator model, or a support packet does not automatically carry a graph‑ring action. The main algebraic results are the standard but essential ones, stated with their exact scope. Free graph modules satisfy the universal property; their ranks are well defined because R is a nonzero commutative ring. The category Modₑ has kernels, cokernels, images, coimages, finite biproducts, enough projectives, and enough injectives. Free modules are projective; projective modules are precisely direct summands of free modules. Tensor products, internal Hom modules, and the tensor‑Hom adjunction are constructed over R, giving the formal basis for Tor^R and Extₑ. Several methodological repairs are made explicit. Vertex number is a ring character, not a grading of the graph ring by graph size. The regular module is R itself, equivalently the free module on one generator, not the free module on all graph isomorphism classes. A fixed graph’s support bag or scanning algebra is usually not closed under scalar multiplication by arbitrary graph classes; one must either pass to a global free graph‑atom module and take the generated submodule, or supply a genuine representation R A. Finally, the naive definition Extⁿₑ (R, F) gives no higher cohomology because R is projective over itself. Nontrivial graph cohomology must therefore be built from a graph‑dependent chain complex or a nontrivial graph quotient module. These repairs turn graph module theory into a theorem‑safe foundation for graph homological algebra rather than a relabeling of ordinary module theory. Keywords graph module · graph ring · Wang framework · support packet · scanning table · abelian category · projective module · injective module · tensor product · homological algebra