On \(z^0\)-ideals in \(C(X)\). (English) Zbl 0991.54014

Azarpanah, F.; Karamzadeh, O. A. S.; Rezai Aliabad, A.

Fundam. Math. 160, No. 1, 15-25 (1999).

Summary: An ideal \(I\) in a commutative ring \(R\) is called a \(\text{z}^0\)-ideal if \(I\) consists of zero divisors and for each \(a\in I\) the intersection of all minimal prime ideals containing \(a\) is contained in \(I\). We characterize topological spaces \(X\) for which z-ideals and \(\text{z}^0\)-ideals coincide in \(C(X)\), or equivalently, the sum of any two ideals consisting entirely of zero divisors consists entirely of zero divisors. Basically disconnected spaces, extremally disconnected and P-spaces are characterized in terms of \(\text{z}^0\)-ideals. Finally, we construct two topological almost P-spaces \(X\) and \(Y\) which are not P-spaces and such that in \(C(X)\) every prime \(\text{z}^0\)-ideal is either a minimal prime ideal or a maximal ideal and in \(C(Y)\) there exists a prime \(\text{z}^0\)-ideal which is neither a minimal prime ideal nor a maximal ideal.

MSC:

54C40	Algebraic properties of function spaces in general topology
46J20	Ideals, maximal ideals, boundaries
13A18	Valuations and their generalizations for commutative rings