## 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 However, this is no longer valid if one drops the assumption that $$R$$ is an algebraically closed field; both implications (i) $$\Rightarrow$$ (ii) and (ii) $$\Rightarrow$$ (i) have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 $$R$$.

### MSC:

 13A99 General commutative ring theory 16S34 Group rings

### Keywords:

group algebra; idempotent; torsion abelian group

Zbl 1003.16015
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### References:

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