Narrow Results

We consider those injective modules that determine every left exact preradical and we call them main injective modules. We construct a main injective module for every ring and we prove some of its properties. In particular we give a characterization, in terms of main injective modules, of rings with a dimension defined by a filtration in the lattice of left exact preradicals. We define also the concept of basic preradical and prove some of its properties. In particular we prove that the class of all basic preradicals is a set, giving a bijective correspondence with the set of all left exact preradicals.

We define the concept of "semicoprime" for preradicals and for submodules and we prove some properties that relate both of them. For any ring we define the ultrasocle preradical as a certain join of maximal semicoprime preradicals. It defines a kind of primary decomposition on modules. We compare the greatest semicoprime preradical, the meet of all unipotent preradicals, the socle preradical, and the ultrasocle preradical. We characterize rings which are finite product of simple rings in terms of some of these preradicals. We study the least semicoprime preradical above any preradical and we prove some of its properties. Using "Amitsur constructions" we define some related operators and prove some of their properties.

In this paper, we consider aspects of the big lattice of preradicals, related to pseudocomplements and supplements. We consider essential preradicals and superfluous preradicals, and we characterize the situation in which all nonzero preradicals are essential as well as the one in which all proper preradicals are superfluous. Also, we give two different proofs of the known fact that the big lattice of preradicals is strongly pseudocomplemented.

This paper is devoted to study the 2-absorbing preradicals and 2-absorbing submodules. We investigate some relations between the concept of 2-absorbing preradicals and 2-absorbing submodules using the lattice structure of preradicals.

In this paper, we define the concept of "co-2-absorbing" for preradicals and submodules. Also, we investigate the relationship between co-2-absorbing preradicals and co-2-absorbing submodules.

In this paper we introduce dual $F$-Baer modules and give a characterization of them. Let $M$ be a module and $F$ a fully invariant submodule of $M$. $M$ is called dual $F$-Baer if for every family of endomorphisms ${\{g_{\alpha }\}}_{\alpha \in I}$ of $M$, ${\sum }_{\alpha \in I}{g}_{\alpha }(F)$ is a direct summand of $M$. We prove that $M$ is dual $F$-Baer if and only if $M=F\oplus N$ for some submodule $N$ of $M$ with $F$ dual Baer. We obtain a positive solution for the Schröder–Bernstein problem for certain dual $F$-Baer modules.

Two obvious classes of quasi-injective modules are those of semisimples and injectives. In this paper, we study rings with no quasi-injective modules other than semisimples and injectives. We prove that such rings fall into three classes of rings, namely, (i) QI-rings, (ii) rings with no middle class, or (iii) rings that decompose into a direct product of a semisimple Artinian ring and a strongly prime ring. Thus, we restrict our attention to only strongly prime rings and consider hereditary Noetherian prime rings to shed some light on this mysterious case. In particular, we prove that among these rings, QIS-rings which are not of type (i) or (ii) above are precisely those hereditary Noetherian prime rings which are idealizer rings from non-simple QI-overrings.

For near-ring ideal mappings ρ₁ and ρ₂, we investigate radical theoretical properties of and the relationship among the class pairs (ρ₁: ρ₂), and (ℛ_ρ2: ℛ_ρ1). Conditions on ρ₁ and ρ₂ are given for a general class pair to form a radical class of various types. These types include the Plotkin and KA-radical varieties. A number of examples are shown to motivate the suitability of the theory of Hoehnke-radicals over KA-radicals when radical pairs of near-rings are studied. In particular, it is shown that forms a KA-radical class, where denotes the class of completely prime near-rings and the class of 3-prime near-rings. This gives another near-ring generalization of the 2-primal ring concept. The theory of radical pairs are also used to show that in general the class of 3-semiprime near-rings is not the semisimple class of the 3-prime radical.

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.