A second almost subidempotent radical for rings. To the memory of Professor OTTÓ VARGA (1909–1969)
✍ Scribed by Ferenc A. Szász
- Publisher
- John Wiley and Sons
- Year
- 1975
- Tongue
- English
- Weight
- 446 KB
- Volume
- 66
- Category
- Article
- ISSN
- 0025-584X
No coin nor oath required. For personal study only.
✦ Synopsis
The purpose of this paper, which can be considered, as a continuation of the author's papers [23] and [24], is to show, with the help of some elementfree methods of C. ROOS [19], that the class S of all &-rings (see Definition 1 of our paper) forms a radical class in the sense of AMITSUR and KUROSH (Theorem 8). This radical is almost subidempotent, but it is not hereditary (Remarks 15(3) and (4)). We give examples for &-rings, and assert equivalent conditions, each of which characterizes the &-rings (Propositions 3 and 4). Every locally nilpotent ring is strongly S-semisimple for this radical S (Theorem 9). On the other hand, every strongly S-semisimple ring is antisiniple (Theorem 11). We mention some particular strongly semisimple rings (Corollary 12( 1) and ( 2) ) , furthermore, we characterize the nilpotent rings among all right artinian rings (Theorem 14) with the help of the S-semisimplicity. The relationship a E S ( a ) = a A + Aa + AaA defines a so-called P-regularity for F ( a ) = S(a). In this way an F-radical rad, A in the sense of B. BROWN and N. H. McCoy [7] can be defined for every ring A such that rad,A 2 S ( A ) , and rad A is a more general radical in the sense