{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T18:52:28Z","timestamp":1776711148663,"version":"3.51.2"},"reference-count":0,"publisher":"Wolfram Research, Inc.","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["ComplexSystems"],"published-print":{"date-parts":[[2026,3,15]]},"abstract":"<jats:p>This paper investigates the structural properties of two-dimensional cellular automata (2DCAs) over rings \nZ\n2\n to \nZ\n9\n, 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 \nI\n\u2208\nZ\nk\nn\n\u2a2f\nn\n be a configuration matrix. For rings in \nS\n1\n=\n{\nZ\n2\n,\nZ\n4\n,\nZ\n8\n}\n, repeated elementwise addition modulo k maps the entries of I to their immediate subrings in the containment hierarchy. For \nS\n2\n=\n{\nZ\n3\n,\nZ\n9\n}\n, 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 \nZ\n2\n to \nZ\n5\n and \nZ\n7\n to \nZ\n9\n, proving that at time step t, rule matrices generate multiple non-overlapping replicas of the initial configuration across each ring and its subrings.\n        Experiments in C++ across various rings and image sizes revealed two key patterns requiring further mathematical explanation. Rings with only trivial subrings\u2014\nZ\n2\n, \nZ\n3\n, \nZ\n5\n and \nZ\n7\n\u2014replicate the initial image at time steps \nt\n=\np\nk\n, where \np\n\u2208\n{\n2\n,\n3\n,\n5\n,\n7\n}\n and \nk\n\u2265\n0\n. In contrast, rings with nontrivial subrings\u2014\nZ\n4\n and \nZ\n8\n\u2014generate multiple replicas at \nt\n=\n2\nk\n, while \nZ\n9\n exhibits replication at \nt\n=\n3\nk\n, producing both the original image and its subrings. The ring \nZ\n6\n, with disjoint subrings and no clear hierarchy, shows no such structured replication.<\/jats:p>","DOI":"10.25088\/complexsystems.35.1.11","type":"journal-article","created":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T18:16:01Z","timestamp":1776708961000},"page":"11-38","source":"Crossref","is-referenced-by-count":0,"title":["Exploring the Containment Hierarchy of Subrings over Rings  \u2124 2  to  \u2124 9  in Two-Dimensional Cellular Automata"],"prefix":"10.25088","volume":"35","author":[{"given":"Velvet","family":"Alexander","sequence":"first","affiliation":[{"name":"School of Advanced Sciences, Vellore Institute of Technology Chennai Campus, Tamil Nadu, India\u2014600127"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ummity Srinivasa","family":"Rao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kedar","family":"Shinde","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jayaram","family":"Balabaskaran","sequence":"additional","affiliation":[{"name":"School of Computer Science and Engineering Vellore Institute of Technology Chennai Campus, Tamil Nadu, India\u2014600127"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"R. J.","family":"Hari","sequence":"additional","affiliation":[{"name":"Indian Institute of Information Technology Kottayam Valavoor P.O, Pala Kottayam, Kerala, India\u2014686635"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Asif","family":"Islam","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Aqdus Bin","family":"Bari","sequence":"additional","affiliation":[{"name":"Zakir Husain College of Engineering and Technology Aligarh Muslim University, Uttar Pradesh, India\u2014202002"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"2184","published-online":{"date-parts":[[2026,3,15]]},"container-title":["Complex Systems"],"original-title":[],"link":[{"URL":"https:\/\/content.wolfram.com\/sites\/13\/2026\/04\/35-1-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T18:16:03Z","timestamp":1776708963000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.complex-systems.com\/abstracts\/v35_i01_a02\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,15]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,3,15]]},"published-print":{"date-parts":[[2026,3,15]]}},"URL":"https:\/\/doi.org\/10.25088\/complexsystems.35.1.11","relation":{},"ISSN":["0891-2513"],"issn-type":[{"value":"0891-2513","type":"print"}],"subject":[],"published":{"date-parts":[[2026,3,15]]}}}