that's a trivial result of Wedderburn-Artin (theorem 5.59) and theorem 7.3.Hi:
In McCoy, The Theory of Rings, 1964 I read:
2. Show that if R is a ring such that J(R) = (0) and the d.c.c. for right ideals holds in R, then R is a regular ring. [J = Jacobson radical, d.c.c. = descending chain condition, R is an arbitrary ring i.e., not necessarily commutative or with unity except it is subject to the two specified conditions.]
The author's definition of a regular ring is
7.2 Definition. Let c be an element of an arbitrary ring R. If there exists an element c' of R such that c = cc'c, c is siad to be a regular element of R. The ring R is said to be a regular ring if each of its elements is regular.
Going to the proof, I find in the book
7.28 Theorem. If the d.c.c. for right ideals holds in the ring R, then N(R) = J(R) = B(R).
Nevermind what the radicals N or B are. But I also have this:
7.27 Theorem. If R is a ring with more than one element, then N(R) = (0) iff R is isomorphic to a subdirect sum of simple rings with unity.
So R, in the exercise, can be considered to be a subdirect sum of simple rings with unity and, to deal with the simplest case, that in which the sum consists of only one ring, let us say R is a simple ring with unity. It would be nice if simple rings with unity were always regular. Or simple ring + unity + dcc = regular. This is as far as I could go. Not a long way indeed. Thanks.