On lattice of Basic Z-Ideals

Author
Yasouj University
Abstract
For an f-ring with bounded inversion property, we show that , the set of all basic z-ideals of , partially ordered by inclusion is a bounded distributive lattice. Also, whenever is a semiprimitive ring, , the set of all basic -ideals of , partially ordered by inclusion is a bounded distributive lattice. Next, for an f-ring with bounded inversion property, we prove that is a complemented lattice and is a semiprimitive ring if and only if is a complemented lattice and is a reduced ring if and only if the base elements for closed sets in the space are open and is semiprimitive if and only if the base elements for closed sets in the space are open and is reduced. As a result, whenever (i.e., the ring of continuous functions), we have is a complemented lattice if and only if is a complemented lattice if and only if is a -space../files/site1/files/71/12.pdf
Keywords

1. Ball R. N, "Walters-Wayland, J., C- and C∗-quotients in pointfree topology", Dissertationes Math. (Rozprawy Mat.) 412 (2002) 1-62. ## 2. Banaschewski B., "The real numbers in pointfree topology", Textos de Mathem´atica (S´eries B) No. 12, Departamento de Mathem´atica da Universidade de Coimbra, Coimbra (1997). ## 3. Dube T., "A note on lattice of z-ideals of f-rings", New York J. Math, 22 (2016) 351-361.## 4. Dube T., Ighedo O., "On z-ideals of point free function rings", Bull. Iran. Math. Soc, 40 (2014) 655-673.## 5. Dube T., "Concerning P-frames, essential P-frames and strongly zerodimensional frames", Algebra Universalis, 69 (2009) 115-138.## 6. Gillman L., Jerison M., "Ring of Continuous Functions", Springer (1976).## 7. Mason G., "z-ideals and prime ideals", J. Algebra, 26 (1973) 280-297.## 8. Suzanne L., "A characterization of f-rings in which the sum of semiprime ideals is semiprime and its consequences", Comm. Algebra, 23 (1995) 5461-5481. ## 9. Varadarajan K., "Clean, almost clean, potent commutative rings", J. Algebra Appl, 6 (2007) 671-685.##