Ideals, duality, and automorphisms of multi-branch finite local rings

Let $ k = \mathbb F_q $, let $ r\ge1 $, and let $ \mathbf n = (n_1, \ldots, n_r) $ with $ n_i\ge2 $. We determine the ideal, annihilator, cyclic-module, and automorphism structures of the finite local $ k $-algebra $ R_{\mathbf n} = k[x_1, \ldots, x_r] \Big/ \big\langle x_i^{n_i}, \, x_ix_j\ (i\ne j)\big\rangle. $ Every proper ideal is represented uniquely by a minimal branch-cutoff vector and an admissible terminal subspace. Transition maps and coordinate saturation then give explicit containment, intersection, and sum formulas and reveal projective and Grassmannian fibers in the ideal lattice. A closed formula counts the ideals for every $ r $ and specializes to the chain and two-branch counts. From the same data, we compute minimal generating sets, a bivariate generator–rank enumerator, and a projective-torus parameterization of principal ideals. We also determine annihilators, double-annihilator closure, its exact defect, and the resulting distributive duality core. For cyclic quotients, this closure is the torsionless reflection, in the sense that the quotient by $ I^{\perp\perp} $ is universal among maps into torsionless modules; it yields a complete classification of torsionless cyclic modules and shows that every nonzero proper cyclic quotient has bidual cokernel dimension $ r^2-1 $. Finally, we determine the full automorphism group together with its tangent, socle, and depth-two actions on ideals. The two-branch rings $ k[u, v]/\langle u^\ell, v^\iota, uv\rangle $ are recovered as the first nontrivial specialization.