A unified approach to some results on one-sided ideals and matrix rings of associative rings

We present a method, which allows to obtain in a uniform and simple way some known and new results on one-sided ideals of rings and on matrix rings. Roughly speaking, it is based on the observation that the left and right principal ideals generated by an element of a ring are ring isomorphic each other modulo their prime radicals. This observation gives ring isomorphisms, modulo the prime radical, of left ideals of rings and some right ideals of some matrix rings, which bring various applications.