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Joe Repka
Department of Mathematics, University of Toronto
Mathematics

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Atom of the Lattice of All N-Radicals Puguh Wahyu Prasetyo; Indah Emilia Wijayanti; Halina France-Jackson; Joe Repka
Journal of the Indonesian Mathematical Society Vol. 32 No. 3 (2026): SEPTEMBER
Publisher : IndoMS

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.22342/jims.v32i3.2016

Abstract

In this paper, we study atomic elements in the lattices of radical classes. A minimal element of the lattice of all \(N\)-radicals (respectively, supernilpotent radicals and special radicals) is called an \(N\)-atom (respectively, a supernilpotent atom and a special atom). We construct a class of \(N\)-atoms generated by prime essential rings and investigate their structural properties. We show that these \(N\)-atoms are closely related to supernilpotent atoms and special atoms, and we clarify the precise relationships among these three types of atoms. In particular, we prove that for a prime essential ring \(R\), the associated radicals \(\widetilde{\mathit{l}}_{R}\) and \(\overline{\mathit{l}}_{R}\) generate, respectively, a special atom and a supernilpotent atom. This result provides a positive answer to an open question concerning which prime essential rings give rise to atomic elements in the lattices of all $N-$ radical, special radicals and supernilpotent radicals.
Co-Authors Halina France-Jackson Indah Emilia Wijayanti Puguh Wahyu Prasetyo