Depth of ideals of C(X)

JOURNAL OF ALGEBRA

卷:528

Article

Azarpanah, F.¹  Esmaeilvandi, D.¹  Salehi, A. R.² 

[1] Shahid Chamran Univ Ahvaz, Dept Math, Ahvaz, Iran

[2] Petr Univ Technol, Dept Sci, Ahvaz, Iran

关键词: Regular sequence;    Depth;    Almost P-space;    Classical ring;    Essential ideal;    z-ideal;   

DOI  :  10.1016/j.jalgebra.2019.03.018

来源: Elsevier

【 摘 要 】

We show that every regular sequence in C(X) has length <= 1. This shows that depth(C(X)) <= 1. We also show that the depth of each maximal ideal of C(X) is either zero or one. In fact we observe that X is an almost P-space if and only if the depth of each maximal ideal of C(X) is zero and X contains at least one non-almost P-point if and only if depth(M) = 1 for each maximal ideal M of C(X). Using this it turns out that for a given topological space X, there are no maximal ideals in C(X) with different depths. Regular sequences are also investigated in the factor rings of C(X) and we observe that such sequences in the factor rings of C(X) modulo principal z-ideals have also length <= 1. Finally, we obtain some topological conditions for which the depth of factor rings of C(X) modulo some closed ideals are zero. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free