Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
38(1), 149 - 159 (1997)

Radicals induced by the Total of Rings

K.I. Beidar, R. Wiegandt

Department of Mathematics, National Cheng Kung University, Tainan 70101, Taiwan, ROC,

Mathematical Institute, Hungarian Academy of Sciences, P.O.Box 127, H-1364 Budapest,

Abstract: The total ${\rm Tot}(A)$ of a ring $A$ has been introduced by Kasch (Partiell invertierbare Homomorphismen und das Total, Algebra Berichte 60, Verlag Reinhard Fischer, München 1988) and it reminds of a radical property, though the shortcoming is that ${\rm Tot}(A)$ need not be an ideal because it is not closed under addition. To overcome this difficulty, the following idea is plausible: the upper radical ${\cal K}$ of all rings $A$ with ${\rm Tot}(A)=0$, may be a decent radical. We call ${\cal K}$ the {\it Kasch radical}, and show that ${\cal K}$ is a supernilpotent normal radical. Simultaneously we consider also the upper radical ${\cal K}_p$ of all prime rings $A$ with ${\rm Tot}(A)=0$. ${\cal K}_p$ is a special normal radical containing the Kasch radical ${\cal K}$. One of the benefits of radical theoretical investigations is to explore rings with interesting but unusual properties. This is featured in this note by constructing a biregular ring $G$ such that ${\rm Tot}(G)=0$ and ${\rm Tot}(G/P)\not=0$ for each prime ideal $P$ of $G$. The existence of such a ring is crucial, it shows that ${\cal K}\not={\cal K}_p$. Thus, in a natural way we have got a nonspecial supernilpotent normal radical, namely ${\cal K}$. We also position the radicals ${\cal K}$ and ${\cal K}_p$ in the lattice of all radicals. The Kasch radical is bigger than the Behrens and the Jacobson radicals, and is incomparable with the Brown--McCoy radical, the generalized nil-radical, and the strongly prime radical. Rings with zero total have been studied by the first author in the forthcoming paper (On rings with zero total, Beiträge Alg. Geom., to appear).

Keywords: supernilpotent, normal and special radicals, total

Classification (MSC91): 16N80

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