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

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 )

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

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

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

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

Proof: Let X be normal. 12

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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

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

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THANKS 25

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

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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

[ 1] C.W. Kohls, Ideals in rings of continuous functions, Fund. Math. 45(1957), [2] C.W. Kohls, Prime ideals in rings of continuous functions, Illinois J. Math. 2(1958), [3] C.W. Kohls, Prime ideals in rings of continuous functions, II, Duke Math. J. 25(1958), 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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[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), [2] T. Shirota, A class of topological spaces, Osaka Math. J. 4(1952), Question: Is the sum of every two closed ideals in C(X) a closed ideal? 32

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

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

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

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[F. Azarpanah and R. Mohamadian ] 40

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

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

# 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

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), F. Azarpanah, S. Afrooz and O. A. S. Karamzadeh, Goldie dimension of rings of fractions of C(X), submitted. 49

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), W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229(1977),

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

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

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

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