{"id":567,"date":"2024-03-10T18:31:08","date_gmt":"2024-03-10T18:31:08","guid":{"rendered":"https:\/\/pri.tum.ac.ke\/?page_id=567"},"modified":"2024-03-10T18:31:08","modified_gmt":"2024-03-10T18:31:08","slug":"ideals-of-rl","status":"publish","type":"page","link":"https:\/\/pri.tum.ac.ke\/?page_id=567","title":{"rendered":"Ideals of RL"},"content":{"rendered":"\n<p>S. NYOKABI, T. DUBE, AND M. MUNYWOKI<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Abstract<\/h3>\n\n\n\n<p>Each frame <em>L<\/em>has associated with it the ring <em>R<\/em><em>L<\/em>= (<em>Frm<\/em>(<em>R<\/em><em>L<\/em>)<em>,<\/em><em>L<\/em>) of its continuous real functions. In this talk, we study ideals of pointfree func- tion rings particularly the <em>z<\/em>-ideals and <em>d<\/em>-ideals of the ring <em>R<\/em><em>L<\/em>of continuous real-valued functions on a completely regular frame <em>L<\/em>. The idea of <em>z<\/em>-ideals came about in the study of the ideal structure of the ring <em>C<\/em>(<em>X<\/em>) of real-valued continuous functions on a completely regular Hausdorff space <em>X<\/em>by C. W. Kohls (1957). In 1971, Gordon Mason initiated the study of <em>z<\/em>-ideals in com- mutative rings with identity. Given that every positive element of <em>R<\/em><em>L<\/em>is a square, we show that an ideal of <em>R<\/em><em>L <\/em>is a <em>z<\/em>-ideal if and only if its radical is a <em>z<\/em>-ideal and that an ideal of <em>R<\/em><em>L<\/em>is a <em>z<\/em>-ideal if and only if every prime ideal minimal over it is a <em>z<\/em>-ideal. To close off, we show <em>M<\/em><sub><em>J<\/em><\/sub>is a maximal ideal if and only if <em>J<\/em>is a prime element of <em>\u03b2L<\/em>by the Gelfand representation borrowing from T. Dube\u2019s proof(2009).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>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 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":113,"menu_order":4,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-567","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=\/wp\/v2\/pages\/567","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=567"}],"version-history":[{"count":1,"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=\/wp\/v2\/pages\/567\/revisions"}],"predecessor-version":[{"id":568,"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=\/wp\/v2\/pages\/567\/revisions\/568"}],"up":[{"embeddable":true,"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=\/wp\/v2\/pages\/113"}],"wp:attachment":[{"href":"https:\/\/pri.tum.ac.ke\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=567"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}