Rings Whose Nilpotent Elements Form a Wedderburn Radical Subring

We introduce and study a class of rings, which we call NRW rings, distinguished by the condition that the set of nilpotent elements forms a Wedderburn radical subring. This class includes symmetric and semicommutative rings, as well as many other well-known and important classes. We also define nilpotent-semicommuting rings as those satisfying the semicommutativity condition restricted to nilpotent elements and prove that every nilpotent-semicommuting ring is an NRW ring. We provide an element-wise characterization of NRW rings and show that the NRW property is symmetric with respect to one-sided principal ideals. Based on the right ideals of a ring that are NRW, for any ring R we inductively construct the ideal W¯(R) and prove that the prime radical of R equals the intersection of W¯(R) with the sum of all nil right ideals of R. As a consequence, a positive solution to the Köthe conjecture follows for all rings R satisfying R=W¯(R). We also characterize when certain classical ring constructions, such as direct sums, matrix rings, and the Dorroh extension, yield an NRW ring.