{"id":3788,"date":"2014-07-19T14:21:27","date_gmt":"2014-07-19T14:21:27","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=3788"},"modified":"2014-07-19T14:21:27","modified_gmt":"2014-07-19T14:21:27","slug":"lemma-of-the-day-33","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3788","title":{"rendered":"Lemma of the day"},"content":{"rendered":"<p>Let R be a ring. Let n > 0. Let M be an R-module generated by < n elements. Let f : R<sup>&oplus; n<\/sup> &#8212;> M be an R-module map. Then Ker(f) is nonzero. See <a href=\"http:\/\/stacks.math.columbia.edu\/tag\/05WI\">Lemma Tag 05WI<\/a>.<\/p>\n<p>Slogan: R^{n + 1} &#8212;> R^n cannot be injective.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let R be a ring. Let n > 0. Let M be an R-module generated by < n elements. Let f : R&oplus; n &#8212;> M be an R-module map. Then Ker(f) is nonzero. See Lemma Tag 05WI. Slogan: R^{n &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3788\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-3788","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3788","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3788"}],"version-history":[{"count":5,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3788\/revisions"}],"predecessor-version":[{"id":3793,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3788\/revisions\/3793"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3788"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3788"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3788"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}