SPECIAL RADICALS FOR NEAR-RING

The concept of a special radical for near-rings has been treated in several nonequivalent, but related, ways in the recent literature. We use the version due to K. Kaarli to establish that various prime radicals and the nil radical are special radicals on the class A of all near-rings which satisfy an extended version of the Andrunakievich Lemma. Since A includes all d.g. near-rings—and much more—these results significantly extend results previously obtained by Kaarli and by Groenewald. We also obtain special radical results for the Jacobson type radicals 30 and 3 1 , albeit on less extensive classes. Examples are given which illustrate and delimit the theory developed.

The introduction of leucotomy in Germany: National Socialism, émigrés, a divided Germany and the development of neurosurgery

Thinking about the chronology of the introduction of leucotomy in Germany sheds new light on the hypothesis of a special ‘radical’ approach of German psychiatry to the treatment of the mentally ill during the period of National Socialism. Moreover, it offers new insights into the transnational and interdisciplinary conditions of the introduction of leucotomy in early divided post-war Germany.

Special, radical, failure of reduction in psychiatry

Use of network models to identify causal structure typically blocks reduction across the sciences. Entanglement of mental processes with environmental and intentional relationships, as Borsboom et al. argue, makes reduction of psychology to neuroscience particularly implausible. However, in psychiatry, a mental disorder can involve no brain disorder at all, even when the former crucially depends on aspects of brain structure. Gambling addiction constitutes an example.

A PRIME ESSENTIAL RING THAT GENERATES A SPECIAL ATOM

A special atom (respectively, supernilpotent atom) is a minimal element of the lattice $\mathbb{S}$ of all special radicals (respectively, a minimal element of the lattice $\mathbb{K}$ of all supernilpotent radicals). A semiprime ring $R$ is called prime essential if every nonzero prime ideal of $R$ has a nonzero intersection with each nonzero two-sided ideal of $R$. We construct a prime essential ring $R$ such that the smallest supernilpotent radical containing $R$ is not a supernilpotent atom but where the smallest special radical containing $R$ is a special atom. This answers a question put by Puczylowski and Roszkowska.

On Jacobson Near-rings and Special Radicals

In this paper, we construct special radicals using class pairs of near-rings. We establish necessary conditions for a class pair to be a special radical class. We then define Jacobson-type near-rings and show that in most cases the class of all near-rings of this type is a special radical class. Subsequently, we investigate the relationship between each Jacobson-type near-ring and the corresponding matrix near-ring.

ON SPECIAL RADICALS COINCIDING ON SIMPLE RINGS AND ON POLYNOMIAL RINGS

Answering a problem of M. Ferrero, we construct a special radical δ such that δ is contained in the Jacobson radical J, δ and J coincides on simple rings and on polynomial rings, but δ ≠ J. For two special radicals with the above conditions, we give a criterion of their coincidence.

Special radicals and matrix near-rings

Special radical classes of near-rings are defined and investigated. It is shown that our approach, which differs from previous ones, does cater for all the well-known radicals of near-rings. Moreover, most of the desirable properties from their ring theory counterpart are retained. The relationship between the special radical of a near-ring and the corresponding matrix near-ring is given.

On rings generating atoms of lattices of special and supernilpotent radicals

This note is to indicate a nonsemiprime ring R such that the smallest supernilpotent (respectively special) radical containing the ring R is an atom of the lattice of all supernilpotent (respectively special) radicals. This gives a positive answer to Puczylowski's and Roszkowska's question.