Atom of the Lattice of All N-Radicals

Résumé fourni par la source

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.

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