Strongly prime radical of group algebras

Kanchan Joshi and R. K. Sharma

Abstract: The strongly prime radical of a ring $R$, $S\msc{P}(R)$, is defined as the intersection of all strongly prime ideals of $R$. For a group algebra $KG$, we show that $S\msc{P}(KG)\cap K[\De(G)]=\msc{P}(K[\De(G)])$. We prove that the concepts of semiprime and semi-strongly prime coincide in case of PI algebras. Given $S$ over $R$ is a finite normalizing extension of rings, we study the relationship of the $*$-prime radicals of $S$ and $R$. Finally, we give examples of group algebras $KG$ for which $S\mathscr{P}(KG)= *$-$\mathscr{P}(KG)$ and $\mathscr{P}(KG) = S\mathscr{P}(KG)$. Also an example of a group $G$ is constructed for which $\msc{P}(KG)\subsetneq *$-$\msc{P}(KG)\subsetneq S\msc{P}(KG)$.

Keywords: strongly prime radical, $*$-prime radical, prime radical, group algebras

Classification (MSC2000): 16S34, 16N60, 20C07

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