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In this paper, we study large classes of nonlinear systems that admit a transfer function completely described by their input-output behavior. Our objective is to identify and analyze the aforementioned classes, which exhibit unique characteristics related to separable systems. We aim to fit certain examples of automata/dynamical systems with new concepts. We observe that for a Discrete System or Automaton, the state set forms a group. Furthermore, there exists a natural near-ring for Separable Systems. Some substructures of this near-ring are generated by id. and a map from state set Q to itself if the state set Q is an abelian structure. It is interesting to note that Separable Systems themselves form a nearring with respect to parallel and series connections. We discuss certain results and provide examples to validate separable systems and the outcomes. This paper offers a theoretical and practical overview of dynamical systems in our daily lives. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with unity and M be an R-module. The aim of this article is to introduce and investigate certain properties of a new notion of prime submodules of a module, namely roughly prime submodules. A submodule N of M is said to be a roughly prime submodule of M if for rm = N, r = R and m = M, implies that either m = N + Z(M) or r = (N + Z(M): M) where Z(M) is the singular submodule of M. The interaction of this submodule with other classes of modules as well as its characteristics in terms of direct sum, intersection and homomorphic image are studied along with the exploration of its behaviour in quotient modules. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with unity. Two new concepts namely "e-small" and "e-supplement" in R are introduced and many of its properties are discussed in this paper. [ABSTRACT FROM AUTHOR]

Automaton is a system that spontaneously gives an output from an input. The input may be energy, information, materials, etc. The system works without the intervention of man. Simply automaton (plural: automata or au-tomatons) is a self{operating machine. Its synonym is ROBOT. In this paper, the authors study the theory of finite automata rings. Finite automata rings extend the notion of finite automata and lead to the study of various properties of rings ob-tained by using finite automata. Also, commutative automata rings, zero-divisors of an automata ring, automata integral domain, sub-automata rings, ideal of an automata ring and related substructures along with automata ring homomorphism are defined. Some results of rings in terms of automata are given in this paper together with their proofs. Besides, some properties of ring homomorphism are derived in terms of automata. Finally, we provide certain examples as well as non-examples of automata rings. [ABSTRACT FROM AUTHOR]