On radicals and extensions in categories (Q1057345)

De la Rosa and Heyman have shown that the radical of the Dorroh extension \(A^*\) of an associative ring A and the radical of A coincide for all rings A if and only if the radical of the ring of integers is zero. This result has been generalized by Feigelstock to arbitrary extensions. This result is true in a much more general setting. In a category with a zero object and a (normal epi, mono)-factorization structure which preserves ideals, the following is true: Let \({\mathcal R}\) be a radical class with the A-D-S property. Then \({\mathcal R}(A)={\mathcal R}(A^*)\) for all objects A if and only if \({\mathcal R}(Y)=0\) where Y is a fixed object in the category such that every object A has an extension \(A^*\) by Y.