1. 1

2. 2

3. Some exceptional properties: 1.Every prime ideal is contained in a unique maximal ideal. 2. Sum of two prime ideals is prime. 3. The prime ideals containing a given prime ideal form a chain. 3

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5. For each space X, there exists a completely regular Hausdorff space Y such that C(X) ≅ C(Y). 5

6. Major Objective? X is connected ⟺ The only idempotents of C(X) are constant functions 0 and 1. 6 Elements of C ( X ), Ideals of C ( X )

7. f ∈C(X) is zerodivisor⟺ int Z( f ) ≠ϕ Every element of C(X) is zerodivisor ⟺ X is an almost P-space Problem. Let X be a metric space and A and B be two closed subset of X. If (A ⋃B)˚≠ϕ, then either A ˚≠ϕ or B ˚≠ϕ. 7

8. Def. A ring R Is said to be beauty if every nonzero member of R is represented by the sum of a zerodivisor and a nonzerodivisor (unit) element. 8

9. 9

10. ♠. Every member of C(X) can be written as a sum of two zerodivisors 10

11. i Theorem. C(X) is clean iff X is strongly zero-dimensional. 11

12. Proof: Let X be normal. 12

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16. 1.Every z-ideal is semiprime. 2.Sum of z-ideals is a z-ideal. 3.Sum of a prime ideal and a z-ideal is a prime z-ideal. 4.Prime ideals minimal over a z-ideal are z-ideals. 5.If all prime ideals minimal over an ideal are z- ideals, then that ideal is also a z-ideal. 6.If a z-ideal contains a prime ideal, then it is a prime ideal. 16

17. Def. An ideal E in a ring R is called essential if it intersects every nonzero ideal nontrivially. 17

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25. THANKS 25

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27. z - ideals E. Hewitt, Rings of real-valued continuous unctions, I, Trans. Amer. Math. Soc. 4(1948), 54-99 27

28. 28

29. 1- Every ideal in C(X) is a z-ideal 2- C(X) is a regular ring 3- X is a P-space (Gillman-Henriksen) Whenever X is compact, then every prime z-ideal is either minimal or maximal if and only if X is the union of a finite number one-point compactification of discrete spaces. (Henriksen, Martinez and Woods) 29

30. [ 1] C.W. Kohls, Ideals in rings of continuous functions, Fund. Math. 45(1957), 28-50 [2] C.W. Kohls, Prime ideals in rings of continuous functions, Illinois J. Math. 2(1958), 505-536. [3] C.W. Kohls, Prime ideals in rings of continuous functions, II, Duke Math. J. 25(1958), 447-458. Properties of z-ideals in C(X): Every z-ideal in C(X) is semi prime. Sum of z-ideals is a z-ideal. (Gillman, Jerison)(Rudd) Sum of two prime ideal is a prime (Kohls) z-ideal or all of C(X). (Mason) Sum of a prime ideal and a z-ideal is a prime z-ideal or all of C(X). (Mason) Prime ideals minimal over a z-ideal is a z-ideal. (Mason) If all prime ideals minimal over an ideal in C(X) are z-ideals, that ideal is also a z-ideal. (Mullero+ Azarpanah, Mohamadian) Prime ideals in C(X) containing a given prime ideal form a chain. (Kohls) If a z-iIdeal in C(X) contains a prime ideal, then it is a prime ideal. (Kohls) 30

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32. [1] L. Gillman, M. Henriksen and M. Jerison, On a theorem of Gelfand and Kolmogoroff concerning maximal ideals in rings of continuous functions, Proc. Amer. Math. Soc. 5(1954), 447-455. [2] T. Shirota, A class of topological spaces, Osaka Math. J. 4(1952), 23-40. Question: Is the sum of every two closed ideals in C(X) a closed ideal? 32

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34. F. Azarpanah and A. Taherifar, Relative z-ideals in C(X), Topology Appl. 156(2009), 1711-1717. So relative z-ideals are also bridges Relative z-ideals rez-ideals 34

35. [1] C.B. Huijsmans and Depagter, on z-ideals and d-ideals in Riesz spaces I, Indag. Math. 42(A83)(1980), 183-195. [2] G. Mason, z-ideals and quotient rings of reduced rings, Math. Japon. 34(6)(1989), 941-956. [3] S. Larson, Sum of semiprime, z and d l-ideals in class of f-rings, Proc. Amer. Math. Sco. 109(4)(1990), 895-901. 35

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38. [1] F. Azarpanah, O. A. S. Karamzadeh and A. Rezaei Aliabad, On ideal consisting entirely zero divisors, Comm. Algebra, 28(2)(2000), 1061-1073. [2] G. Mason, Prime ideals and quotient of reduced rings, Math. Japon. 34(6)(1989), 941-956. [3] F. Azarpanah and M. Karavan, On nonregular ideals and z0–ideals in C(X), Cech. Math. J. 55(130)(2005), 397-407. 38

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40. [F. Azarpanah and R. Mohamadian ] 40

41. Questions: [3] F. Azarpanah and M. Karavan, On nonregular ideals and z0– ideals in C(X), Cech. Math. J. 55(130)(2005), 397-407. Let X be a quasi space: 41

42. Essential (large) ideals Uniform (Minimal) ideals The Socle of C(X) Socle of R = S(R) = Intersection of essential ideals = Sum of uniform ideals of R 42

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45. Fact: (a) The socle of C(X) is essential iff the set of isolated points of X is dense in X. (b) Every intersection of essential ideals of C(X) is essential iff the set of isolated points of X is dense in X. * When is the socle of C(X) an essential ideal? 45

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47. [1] F. Azarpanah, Essential ideals in C(X), Period. Math. Hungar., 31(2)(1995), 105-112. [2] F. Azarpanah, Intersection of essential ideals in C(X), Proc. Amer. Math. Soc. 125(1997), 2149-2154. [3] O. A. S. Karamzadeh and M. Rostami, On intrinsic topology and some related ideals of C(X), Proc. Amer. Math. Soc. 93(1985), 73-84. 47

48. # Every factor ring of C(X) modulo a principal ideal contains a nonessential prime ideal iff X is an almost P-space with a dense set of isolated points. 48

49. Clean elements Clean ideals An element of a ring R is called clean if it is the sum of a unit and an idempotent. A subset S of R is called clean if each element of S is clean. F. Azarpanah, O. A. S. Karamzadeh and S. Rahmati, C(X) vs. C(X) modulo its socle, Colloquium Math. 111(2)(2008), 315-365. F. Azarpanah, S. Afrooz and O. A. S. Karamzadeh, Goldie dimension of rings of fractions of C(X), submitted. 49

50. C(X) is clean iff C(X) is an exchange ring. R. B. Warfield, A krull-Scmidt theorem for infinite sum of modules, Proc. Amer. Math. Soc. 22(1969), 460-465. W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229(1977), 269-278. 50

51. X is strongly zero dimensional if every functionally open cover of X has an open refinement with disjoint members. 51

52. Th. The following statements are equivalent: 1. C(X) is a clean ring. 2. C*(X) is a clean ring. 3. The set of clean elements of C(X) is a subring. 4. X is strongly zero-dimensional. 5. Every zerodivisor element is clean. 6. C(X) has a clean prime ideal. 52

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54. F. Azarpanah, When is C(X) a clean ring? Acta Math. Hungar. 94(1-2)(2002), 53-58 54

55. J. Martinez and E. R. Zenk, Yosida frames, J. pure Appl. Algebra, 204(2006), 473-492. 55

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