zc-ideals and prime ideals in the ring RcL | Zendy

Ali Akbar Estaji | Zendy; Karimi Feizabadi | Zendy; Robat Sarpoushi | Zendy

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zc-ideals and prime ideals in the ring RcL

Author(s) -

Ali Akbar Estaji,

Karimi Feizabadi,

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.

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