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\(*\)-prime and strongly prime radicals of group algebras. (English) Zbl 1149.16022

Jain, S.K. (ed.) et al., Noncommutative rings, group rings, diagram algebras and their applications. International conference, University of Madras, Chennai, India, December 18–22, 2006. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4285-0/pbk). Contemporary Mathematics 456, 19-26 (2008).
All algebras in this paper are associative algebras with identity \(1\neq 0\) over a field \(K\). Let \(G\) be a group. If \(H\leqq G\), then \(|G:H|=l.f.\) if \(|L:L\cap H|<\infty \) for every finitely generated subgroup \(L\) of \(G\). Let \(\Lambda(G)=\{x\in G:|G:C_G(x)|=l.f.\}\). For \(k\geqq 2\), define \(\Lambda_k(G)\) as follows: let \(\Lambda_1(G)=\Lambda(G)\) and \(\Lambda_k(G)/\Lambda_{k-1}(G)=\Lambda(G/\Lambda_{k-1}(G))\).
An algebra \(R\) is called right strongly prime if for every \(0\neq r\in R\) there exists a finite subset \(X\) of \(R\) such that \(rXt=0\) implies \(t=0\). The right strongly prime radical of \(R\) is defined as \(S\mathcal P(R)=\bigcap\{P\trianglelefteq R:R/P\) is right strongly prime}. An algebra \(R\) is right \(*\)-prime if for every \(0\neq r\in R\) there exists a finitely generated subalgebra \(S\) of \(R\) such that \(rSt=0\) implies \(t=0\). The right \(*\)-prime radical of \(R\) is defined as \(*\)-\(\mathcal P(R)=\bigcap\{P\trianglelefteq R:R/P\) is right \(*\)-prime}. For an algebra \(R\), \(J(R) \) denotes the Jacobson radical of \(R\) and \(N^*(R)=\{r\in R:rS\) is nilpotent for every finitely generated subalgebra \(S\) of \(R\}\).
In this paper the authors study \(*\)-prime and strongly prime algebras and the corresponding radicals. They show that if \(G=\Lambda_n(G) \) for some \(n\), then \(I\cap K[\Lambda(G)]\neq (0)\) for every nonzero ideal \(I\) of the group algebra \(KG\). Then they use this result to show that for such groups, \(KG\) is \(*\)-prime if and only if it is strongly prime. They also show that \(J(KG)=N^*(KG)\) when \(G=\Lambda_n(G)\). Moreover, they show that for this class of groups, \(KG\) is strongly prime if and only if \(L(G)=(1)\), where \(L(G) \) is the unique maximal locally finite normal subgroup of \(G\).
They study the strongly prime and \(*\)-prime radicals of group algebras. They obtain various relationships among the radicals of \(KH\) and \(KG\) when \(H\leqq G\). They call an algebra \(R\) semi strongly prime if its strongly prime radical is zero. They show that when either \(G\) is a residually finite group and \(\text{char\,}K=0\) or \(G\) is residually a finite \(p'\)-group and \(\text{char\,}K=p>0\), then \(KG\) is semi strongly prime. They also give an example of an algebra for which the right and left \(*\)-prime radicals are distinct. Moreover, they prove that if \(H\) is a central subgroup of \(G\) with \(G/H\) RO, then \(*\)-\(\mathcal P(KG)=*\)-\(\mathcal P(KH).KG\).
For the entire collection see [Zbl 1139.16002].

MSC:

16S34 Group rings
16N60 Prime and semiprime associative rings
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
16N80 General radicals and associative rings
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