Nilpotent Elements and Idempotents in Commutative Group Rings#

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Let R be a commutative ring with identity and L an extension of R. Let α ∈ L and denote by R[α] the smallest subring of L containing R and α. In this article, we consider only extensions of kind R[α], when α is an algebraic element over R and the minimal polynomial f(x) of α is monic. In this direction, the following two problems arise in a natural way: i. If R does not contain nilpotent elements, does it follow that this property holds for R[α]? ii. If R does not have nontrivial idempotents, does it follow that it is fulfilled for R[α]. We give a positive answer to these two problems under some conditions of the polynomial f(x).