On normal radicals, N-radicals, and A-radicals

Abstract

In this paper we work in the class of associative rings. Fundamental definitions and properties of radicals may be found in Divinsky [3] and Wiegandt [I 71. The concept of normal radical was introduced by Jaegermann [7] and further properties were given by Sands [12]. l’he concept of A’-radical was introduced by Sands [I I]. In [12] it was pointed out that K-radicals are precisely the supernilpotent normal radicals. Another family of radicals has been introduced bv Gardner [4], who called them -l-radicals. These are the radicals nhich depend only on the additive structure of the ring and they also al-e normal r-acsifv hereditarl. normal radicals. ‘I’he concept of a special radical was introduced by -Indrunakic\.ic [2]. In the nest section we show that certain constructions starting from a normal radical lead to a special normal radical which, since a special radical is supernilpotcnt. must be an N-radical. \Ve give, however, an example to show that an .V-radical need not be special.