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This article embodies a ring theoretic property that preserves the reversibility at non zero k-potents. The structure of k-potents in k-reversible and pseudo k-reversible rings are studied in connection with various kinds of ring extensions. A ring R is said to be k-reversible if and only if ab ∈ K(R) for a, b ∈ R implies ab = ba; and a ring R is called pseudo k-reversible if 0 ≠ ab ∈ K(R) for a, b ∈ K(R) implies ab = ba, where K(R) is the set of all k-potents in R. It is proved that k-reversible rings are pseudo k-reversible but the converse is not true in general. Under certain conditions the polynomial rings of pseudo k-reversible inherit the character of pseudo k-reversible rings. [ABSTRACT FROM AUTHOR]

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 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 class of submodules namely e-secondary submodules. A submodule N of M is said to be e-secondary if the endomorphism given by multiplication by a, a ∈ R, that is, f : N → N such that f(N) = aN, then either aN ≤e M or atN = 0, for some t ∈ N . This notion is then further extended to introduce fully e-secondary submodules of a module. [ABSTRACT FROM AUTHOR]

Let M be a left R−module. In this paper, various results pertaining to the equivalence of discrete modules and pseudo-discrete modules are established. Furthermore, a new class of module termed as (CD)2-module is introduced and certain characteristics of this new class of module, along with its relation with the above mentioned equivalence of discrete modules and pseudo- discrete modules are studied. [ABSTRACT FROM AUTHOR]

A ring R is defined to be a weakly tri normal ring if for all a ∈ R and a tripotent element t of R such that at = 0, implies that Rtra is a nil left ideal of R. It is proved that a ring R is weakly tri normal if and only if tR(1 + t 2 ) ⊆ N∗ (R), where t is an op-tripotent element of R. Furthermore, we introduce the concept of quasi tri normal ring. A ring R is a quasi tri normal ring if at = 0, implies taRt = 0, where t is a tripotent element of R. It is observed that for a weakly tri normal ring R, (1) Every quasi tri normal ring is a weakly tri normal ring, (2) Every weakly tri normal ring is directly finite, (3) If R is a weakly tri normal π-regular ring then R is strongly π-regular. [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]