Open Accesszc-ideals and prime ideals in the ring RcL
Author(s) -
Ali Akbar Estaji,
Abolghasem Karimi Feizabadi,
Maryam Robat Sarpoushi
Publication year - 2018
Publication title -
filomat
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 34
eISSN - 2406-0933
pISSN - 0354-5180
DOI - 10.2298/fil1819741e
Subject(s) - mathematics , subring , associated prime , prime (order theory) , maximal ideal , ideal (ethics) , ring (chemistry) , semiprime ring , prime ideal , countable set , minimal ideal , primitive ring , principal ideal ring , pure mathematics , combinatorics , discrete mathematics , commutative ring , philosophy , chemistry , organic chemistry , epistemology , commutative property
The ring RcL is introduced as a sub-f-ring of RL as a pointfree analogue to the subring Cc(X) of C(X) consisting of elements with the countable image. We introduce zc-ideals in RcL and study their properties. We prove that for any frame L, there exists a space X such that ?L ? OX with Cc(X) ? Rc(OX) ? Rc?L ? R*cL, and from this, we conclude that if ?,? ? RcL, |?| ? |?|q for some q > 1, then ? is a multiple of ? in RcL. Also, we show that IJ = I ? J whenever I and J are zc-ideals. In particular, we prove that an ideal of RcL is a zc-ideal if and only if it is a z-ideals. In addition, we study the relation between zc-ideals and prime ideals in RcL. Finally, we prove that RcL is a Gelfand ring.