An ascending chain condition on Wedderburn radicals

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.