In [1], [2] and [3] no chain conditions occur anywhere; the strongest finiteness condition which had to be postulated for certain parts of the theory was that the rings have nilpotent radicals. (Square brackets refer to the references; [1], [2] and [3] will again be referred to as CPI, CPII and CPIII.) In the present paper we apply the results of CPI, II, III to completely primary rings with chain conditions. See the introduction of CPI for a short introduction to the present paper. All notations and terminologies are the same as for CPI, II, III. We remind the reader of the most important ones. Ring always means commutative ring with unit element 1 and zero element 0; otherwise, ring elements are denoted by lower case Greek letters, while a lower case German letter always denotes an ideal. The radical of a ring A, that is the ideal of its nilpotent elements, is denoted by N(A). The residue class ring A/N(A) is denoted by A and if N(A) has a finite exponent, this exponent is also called the exponent of A. If A C B are two rings where a C A and b C B, the extension of a in B is denoted by a* and the contraction of b in A by b* . (See CPI, sections 6a and 8 for the notions of extension and contraction of ideals.) If a = (a*)* for all a C A, we say that the (e, c)-condition is satisfied in the ring extension A C B. (See CPIII, Section 5, where the (e, c)-condition was formulated only for completely primary rings.) The quotient of two ideals a and a' is denoted by a: a'. An element of a ring which is not a divisor of zero, is called a regular element. The symbols C, <, -+ denote respectively inclusion, proper inclusion and implication.