Nilpotent Elements and Idempotents in Commutative Group Rings^#

^#Communicated by A. Facchini.

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:

1. If R does not contain nilpotent elements, does it follow that this property holds for R[α]?

2. 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).

Keywords:

• Discriminant
• Extension of commutative ring
• Idempotent
• Nilpotent element
• Normal extension

AMS Mathematics Subject Classification (2000):

• 13B25
• 13B02
• 16N40

ACKNOWLEDGMENTS

The research was partially supported by the fund “NI” of Plovdiv University.

The author would like to thank the referee for the valuable suggestions.

Notes

^#Communicated by A. Facchini.