Alireza Olfati
On atomic ideals in some factor rings of C(X,\Bbb Z)

Comment.Math.Univ.Carolin. 62,2 (2021) 259-263.

Abstract: A nonzero R-module M is atomic if for each two nonzero elements a, b in M, both cyclic submodules Ra and Rb have nonzero isomorphic submodules. In this article it is shown that for an infinite P-space X, the factor rings C(X,\Bbb{Z})/C_F(X,\Bbb{Z}) and C_c(X)/C_F(X) have no atomic ideals. This fact generalizes a result published in paper by A. Mozaffarikhah, E. Momtahan, A. R. Olfati and S. Safaeeyan (2020), which says that for an infinite set X, the factor ring \Bbb{Z}^X/ \Bbb{Z}^{(X)} has no atomic ideal. Another result is that for each infinite P-space X, the socle of the factor ring C_c(X)/C_F(X) is always equal to zero. Also, zero-dimensional spaces X are characterized for which C^F(X,\Bbb{Z})/C_F(X,\Bbb{Z}) have atomic ideals.

Keywords: P-space; rings of integer-valued continuous functions; functionally countable subalgebra; atomic ideal; socle

DOI: DOI 10.14712/1213-7243.2021.013
AMS Subject Classification: 54C40