Commutative rings over which algebras generated by idempotents are quotients of group algebras. (English) Zbl 1333.13008

Let \(R\) be a commutative ring with identity \(1_R\) and let \( G\) be an abelian group. \(RG\) denotes the group algebra of \(G\) over \(R\). For an \(R\)-algebra \(A\) consider the following two conditions. (i) \(A\) is generated by idempotents over \(R\); (ii) \(A\) is a quotient ring of \(RG\) for a torsion abelian group \(G\) without an element whose order is a zero-divisor in \(R\).
In [H. Kawai, Commun. Algebra 30, No. 1, 119–128 (2002; Zbl 1003.16015)], it has been shown that, if \(R\) is an algebraically closed field, then the above two conditions are equivalent. However, this is no longer valid if one drops the assumption that \(R\) is an algebraically closed field; both implications (i) \(\Rightarrow\) (ii) and (ii) \(\Rightarrow\) (i) have counter-examples even if \(R\) is a field.
In this paper the authors study the relationship between algebras generated by idempotents over a commutative ring \(R\) with identity and algebras that are quotient rings of group algebras \(RG\) for torsion abelian groups \(G\) without an element whose order is a zero-divisor in \(R\).


13A99 General commutative ring theory
16S34 Group rings


Zbl 1003.16015
Full Text: DOI Euclid


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