Narrow Results

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.

A Noetherian (Artinian) Lie algebra satisfies the maximal (minimal) condition for ideals. Generalisations include quasi-Noetherian and quasi-Artinian Lie algebras. We study conditions on prime ideals relating these properties. We prove that the radical of any ideal of a quasi-Artinian Lie algebra is the intersection of finitely many prime ideals, and an ideally finite Lie algebra is quasi-Noetherian if and only if it is quasi-Artinian. Both properties are equivalent to soluble-by-finite. We also prove a structure theorem for serially finite Artinian Lie algebras.

This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings. The properties of radicals of pseudo-reduced-over-center rings are investigated, especially related to polynomial rings. It is proved that for pseudo-reduced-over-center rings of nonzero characteristic, the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals. For a locally finite ring $R$, it is proved that if $R$ is pseudo-reduced-over-center, then $R$ is commutative and $R/J(R)$ is a commutative regular ring with $J(R)$ nil, where $J(R)$ is the Jacobson radical of $R$.

We study the structure of rings which satisfy the von Neumann regularity of commutators, and call a ring $R$C-regular if $ab-ba\in (ab-ba)R(ab-ba)$ for all $a$, $b$ in $R$. For a C-regular ring $R$, we prove $J(R[X])=N^{\ast }(R[X])=N^{\ast }(R)[X]=W(R)[X]\subseteq Z(R[X])$, where $J(A)$, $N^{\ast }(A)$, $W(A)$, $Z(A)$ are the Jacobson radical, upper nilradical, Wedderburn radical, and center of a given ring $A$, respectively, and $A[X]$ denotes the polynomial ring with a set $X$ of commuting indeterminates over $A$; we also prove that $R$ is semiprime if and only if the right (left) singular ideal of $R$ is zero. We provide methods to construct C-regular rings which are neither commutative nor von Neumann regular, from any given ring. Moreover, for a C-regular ring $R$, the following are proved to be equivalent: (i) $R$ is Abelian; (ii) every prime factor ring of $R$ is a duo domain; (iii) $R$ is quasi-duo; and (iv) $R/W(R)$ is reduced.