**
Beiträge zur Algebra und Geometrie **

Contributions to Algebra and Geometry

38(1), 149 - 159 (1997)

#
Radicals induced by the Total of Rings

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K.I. Beidar, R. Wiegandt

Department of Mathematics, National Cheng Kung University, Tainan 70101, Taiwan, ROC, t14270@sparc1.cc.ncku.edu.tw Mathematical Institute, Hungarian Academy of Sciences, P.O.Box 127, H-1364 Budapest, wiegandt@math-inst.hu

**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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