{"id":474,"date":"2025-07-18T09:50:11","date_gmt":"2025-07-18T08:50:11","guid":{"rendered":"https:\/\/mat.uab.cat\/web\/tfg\/?p=474"},"modified":"2025-09-08T20:28:27","modified_gmt":"2025-09-08T19:28:27","slug":"commutadors-en-anells-i-algebres","status":"publish","type":"post","link":"https:\/\/mat.uab.cat\/web\/tfg\/commutadors-en-anells-i-algebres\/","title":{"rendered":"Commutadors en anells i \u00e0lgebres"},"content":{"rendered":"\n<p>El commutador additiu de \\(x\\) i \\(y\\) en un anell \\(R\\) \u00e9s per definici\u00f3 \\([x,y]=xy-yx\\), que \u00e9s trivial quan l&#8217;anell \u00e9s commutatiu, i per tant adquireix import\u00e0ncia en el cas no commutatiu (per exemple, en anells de matrius sobre un cos). En el treball es pret\u00e9n estudiar anells molt no commutatius, en el sentit que els commutadors els generen com a ideal. Els primers resultats en aquesta direcci\u00f3 es remunten a l&#8217;any 65 i han tingut un resorgiment recent, al 2024. En particular \u00e9s interessant estudiar, donat \\(R\\), el m\u00ednim \\(N\\) tal que tot element \\(a\\in R\\) s&#8217;escriu com \\(a=\\sum_{i=1}^N[a_i,b_i][c_i,d_i]\\). Aquest \\(N\\) es denota per \\(\\xi(R)\\) i ha estat estimat en algunes situacions (per anellls purament algebraics i tamb\u00e9 per \u00e0lgebres topol\u00f2giques). S&#8217;estudiar\u00e0 el cas del comportament d&#8217;aquest invariant pel pas a matrius amb especial \u00e8mfasi a \\(M_n(D)\\), on \\(D\\) \u00e9s un cos (no necess\u00e0riament commutatiu) i la relaci\u00f3 amb l&#8217;anomenada conjectura de L&#8217;vov-Kaplansky. Tamb\u00e9 s&#8217;abordar\u00e0 en el cas d&#8217;\u00e0lgebres d&#8217;operadors, si el temps ho permet.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>El commutador additiu de \\(x\\) i \\(y\\) en un anell \\(R\\) \u00e9s per definici\u00f3 \\([x,y]=xy-yx\\), que \u00e9s trivial quan l&#8217;anell \u00e9s commutatiu, i per tant adquireix import\u00e0ncia en el cas no commutatiu (per exemple, en anells de matrius sobre un cos). En el treball es pret\u00e9n estudiar anells molt no commutatius, en el sentit que [&hellip;]<\/p>\n","protected":false},"author":22,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[28,1],"tags":[46],"class_list":["post-474","post","type-post","status-publish","format-standard","hentry","category-algebra","category-general","tag-francesc-perera"],"_links":{"self":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/474","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/users\/22"}],"replies":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/comments?post=474"}],"version-history":[{"count":2,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/474\/revisions"}],"predecessor-version":[{"id":476,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/474\/revisions\/476"}],"wp:attachment":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/media?parent=474"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/categories?post=474"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/tags?post=474"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}