S. NYOKABI, T. DUBE, AND M. MUNYWOKI
Abstract
Each frame Lhas associated with it the ring RL= (Frm(RL),L) of its continuous real functions. In this talk, we study ideals of pointfree func- tion rings particularly the z-ideals and d-ideals of the ring RLof continuous real-valued functions on a completely regular frame L. The idea of z-ideals came about in the study of the ideal structure of the ring C(X) of real-valued continuous functions on a completely regular Hausdorff space Xby C. W. Kohls (1957). In 1971, Gordon Mason initiated the study of z-ideals in com- mutative rings with identity. Given that every positive element of RLis a square, we show that an ideal of RL is a z-ideal if and only if its radical is a z-ideal and that an ideal of RLis a z-ideal if and only if every prime ideal minimal over it is a z-ideal. To close off, we show MJis a maximal ideal if and only if Jis a prime element of βLby the Gelfand representation borrowing from T. Dube’s proof(2009).