Lattices of Radicals of Ω-Groups

Author:

Booth G. L.1,Petersen Q. N.1,Veldsman S.2

Affiliation:

1. Department of Mathematics, University of Port Elizabeth, P.O. Box 1600, Port Elizabeth 6000, South Africa

2. Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khodh PC123, Sultanate of Oman

Abstract

Snider initiated the study of lattices of the class of radicals, in the sense of Kurosh and Amitsur, of associative rings. Various authors continued the investigation in more general universal classes. Recently, Fernández-Alonso et al. studied the lattice of all preradicals in R-Mod. Our definition of a preradical is weaker than theirs. In this paper, we consider the lattices of ideal maps 𝕀, preradical maps ℙ, Hoehnke radical maps ℍ and Plotkin radical maps 𝔹 in any universal class of Ω-groups (of the same type). We show that 𝕀 is a complete and modular lattice which contains atoms. In general, 𝕀 is not atomic. 𝕀 contains ℙ as a complete and atomic sublattice, whereas ℍ and 𝔹 are not sublattices of 𝕀. In its own right, ℍ is a complete and atomic lattice and 𝔹 is a complete lattice. We identify subclasses of 𝕀, ℙ and ℍ that are sublattices or preserve the meet (or join) of these respective lattices.

Publisher

World Scientific Pub Co Pte Lt

Subject

Applied Mathematics,Algebra and Number Theory

Reference22 articles.

1. Lecture Notes in Pure and Appl. Math., 75;Bican L.,1982

2. LATTICES OF RADICALS OF NEAR-RINGS

3. A Course in Universal Algebra

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