On Topological Ring(創刊10周年記念号)

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　In　this　territory，E．Noether，W．Krull，C．Chevalley and　I．S．Cohen　established many　important results．　More　recently，O．Zariski　gave　me　honour to　get　his　well-known writings　in　this　aspects． He　introduced　metrics　into　a　ring　as　§l．We　shall　extend　some　of　his　ideas．　We　shall　treatize，especially，　a　semi－local　ring，　because　this　ring　has　a　natural　topology，　namely，　determined　by　the　ideal　which　is　the　intersection　or，　what　is　the　same　thing，　the　product　of　the　maximal　ideals．We　have　tried　to　apply　a　topology　to　a　ring　as　far　as　possible．　We　can　construct　the　completion　of　the　ring，　the　idea　of　the　closed　ideal，　closure　and　Cauchy　sequence，　and　also　compactness　and　connectivity　of　a　ring，　with　respect　to　this　topoiogy．　Some　results　are　denoted　below，　but　theconnectivity　of　a　ring　will　be　written　in　my following　paper．
滋賀大学の論文