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**Abstract**

Let H = {R | R is a commutative ring and Nil(R) is a divided prime ideal of R}. For a ring R in H with total quotient ring T(R), Let phi be the natural ring homomorphism from T(R) into R_Nil(R). An integral domain R is said to be an FC-domain (in the sense of Gilmer) if each chain of distinct overrings of R is finite, and R is called an FO-domain if R has finitely many overrings. A ring R is called an FC-ring if each chain of distinct overrings of R is finite, and R is said to be an FO-ring if R has finitely many overrings. A ring R in H is said to be a phi-FC-ring if phi(R) is an FC-ring, and R is called a phi-FO-ring if phi(R) is an FO-ring. In this paper, we show that the theory of phi-FC-rings and phi-FO-rings resembles that of FC-domains and FO-domains.