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In this paper, we extend our investigation about the generalized comaximal graph introduced in Biswas et al. (Discrete Math Algorithms Appl 11(1):1950013, 2019a). The generalized comaximal graph is defined as follows: given a finite commutative ring R, the generalized comaximal graph G(R) is an undirected graph with its vertex set comprising elements of R and two distinct vertices u, v are adjacent if and only if there exists a non-zero idempotent e ∈ R such that u R + v R = e R . In this study, we focus on identifying the rings R for which the graph G(R) exhibits planarity. Moreover, we provide a characterization of the class of ring for which G(R) is toroidal, denoted by γ (G (R)) = 1 . Furthermore, we also evaluate the energy of the graph G(R). Finally, we demonstrate that the graph G(R) is always Hamiltonian for any finite commutative ring R. [ABSTRACT FROM AUTHOR]

We fill a gap in the proof of one of the central theorems in Epp’s paper, concerning p-cyclic extensions of complete discrete valuation rings. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with 1. In Biswas et al. (Disc Math Algorithms Appl 11(1):1950013, 2019), we introduced a graph G(R) whose vertices are elements of R and two distinct vertices a, b are adjacent if and only if a R + b R = e R for some nonzero idempotent e in R. Let G ′ (R) be the subgraph of G(R) generated by the non-units of R. In this paper, we characterize those rings R for which the graph G ′ (R) is connected and Eulerian. Also we characterize those rings R for which genus of the graph G ′ (R) is ≤ 2 . Finally, we show that the graph G ′ (R) is a line graph of some graph if and only if R is either a regular ring or a local ring. [ABSTRACT FROM AUTHOR]

The paper adjoins the work of B. I. Plotkin and S. M. Vovsi, Varieties of Representations of Groups (Zinatne, Riga (1983)), and turns out to be, in a sense, its continuation. In the former, the varieties of representations have been considered. As a matter of fact, the varieties under consideration are action-type varieties. This paper studies other classes of representations, axiomatizable in a special action-type logic. [ABSTRACT FROM AUTHOR]

Let R be an arbitrary finite commutative local ring. In this paper, we obtain a necessary and sufficient condition for a function over R to be a polynomial function. Before this paper, necessary and sufficient conditions for a function to be a polynomial function over some special finite commutative local rings were obtained. [ABSTRACT FROM AUTHOR]

Abstract.  In this paper we establish several equivalent conditions for every regular element (quasi-regular ideal) of a principally generated C-lattice L (a commutative ring with identity R) to be an invertible element of L (quasi-invertible ideal of R). Using these results, some new characterizations are given for almost Dedekind lattices, almost Dedekind rings, Dedekind lattices and Dedekind rings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a notion of weak Fountain-Gould left order for associative pairs and give a Goldie-like theory of associative pairs which are weak Fountain-Gould left orders in semiprime pairs coinciding with their socles. [ABSTRACT FROM AUTHOR]

A class Κ of rings has the GADS property (i.e., generalized ADS property) if whenever X [This symbol cannot be presented in ASCII format] I [This symbol cannot be presented in ASCII format] R. with X ∈ Κ, then there exists B [This symbol cannot be presented in ASCII format] R with B ∈ Κ such that X [This symbol cannot be presented in ASCII format] B [This symbol cannot be presented in ASCII format] I. Radicals whose semisimple classes have the GADS property axe called g-radicals. In this paper, we fully characterize the class of g-radicals. We show that p is a g-radical if and only if either ρ [This symbol cannot be presented in ASCII format]or Sρ [This symbol cannot be presented in ASCII format] I, where I denotes the class of idempotent rings and Sρ denotes the semisimple class of ρ. It is also shown that the (hereditary) g-radicals form an (atomic) sublattice of the lattice of all radicals. [ABSTRACT FROM AUTHOR]