An Ascending Chain Condition on Wedderburn Radicals

ABSTRACT

In this note we study the properties of Amitsur's example for Wedderburn radicals, introducing the concept of W _n -reduced rings. The theories of commutative ring and reduced ring are generalized to W _n -reduced rings. We characterize the W _n -reduced property and study properties of W _n -reduced rings. It is shown that the classes of semi-commutative rings, W _n -reduced rings, and 2-primal rings are in a strictly increasing order. We extend the class of W _n -reduced rings, observing various kinds of extensions containing classical quotient rings, polynomial rings, and power series rings.

Communicated by M. Ferrero.

Key Words:

• Amitsur's example for Wedderburn radical
• Classical quotient ring
• Polynomial ring
• Wedderburn radical
• W _n -reduced ring

2000 Mathematics Subject Classification:

• 16D25
• 16N40
• 16S36
• 16U20

ACKNOWLEDGMENTS

The authors are indebted to the referee for various valuable comments leading to improvements of the article. The first and third named authors were supported by the Korea Research Foundation Grant (KRF-2001-015-DP0005), while the second named author was supported by the Korea Research Foundation Grant (R05-2004-000-10212-0).