Exploring the Containment Hierarchy of Subrings over Rings Z2 to Z9 in Two-Dimensional Cellular Automata

This paper investigates the structural properties of two-dimensional cellular automata (2DCAs) over rings 12 to 19, focusing on rule composition, subring hierarchy and linear evolution. We prove that rule composition is commutative across all rings and that matrix transformations preserve hierarchical relationships. The subring containment is determined by the divisors of n: prime rings have only trivial subrings, while composite rings exhibit structured hierarchies. Let I E 1knxn be a configuration matrix. For rings in 61 = {12, 14, 18}, repeated elementwise addition modulo k maps the entries of I to their immediate subrings in the containment hierarchy. For 62 = {13, 19}, triple summation modulo k similarly restricts the values to their corresponding subrings. Furthermore, we extend Moore neighborhood-based two-dimensional cellular automaton (2DCA) rules to rings from 12 to 15 and 17 to 19, proving that at time step t, rule matrices generate multiple non-overlapping replicas of the initial configuration across each ring and its subrings. Experiments in C++ across various rings and image sizes revealed two key patterns requiring further mathematical explanation. Rings with only trivial subrings-12, 13, 15 and 17-replicate the initial image at time steps t = pk, where p E {2, 3, 5, 7} and k >= 0. In contrast, rings with nontrivial subrings-14 and 18-generate multiple replicas at t = 2k, while 19 exhibits replication at t = 3k, producing both the original image and its subrings. The ring 16, with disjoint sub-rings and no clear hierarchy, shows no such structured replication.