## Some finiteness conditions on the set of overrings of a phi-ring

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**Date**

2008#####
**Author**

Badawi, Ayman

Jaballah, Ali

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**Advisor(s)**

Unknown advisor#####
**Type**

Article

Published version

Peer-Reviewed

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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.#####
**DSpace URI**

http://hdl.handle.net/11073/9225
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**External URI**

http://www.math.uh.edu/~hjm/Vol34-2.html
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