{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T11:21:27Z","timestamp":1770895287775,"version":"3.50.1"},"reference-count":2,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":14256,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1975,3]]},"abstract":"<jats:p>In [1] it was claimed that the word problem for free fields with infinite centre can be solved. In fact it was asserted that if <jats:italic>K<\/jats:italic> is a skew field with infinite central subfield <jats:italic>C<\/jats:italic>, then the word problem in the free field on a set <jats:italic>X<\/jats:italic> over <jats:italic>K<\/jats:italic> can be solved, relative to the word problem in <jats:italic>K<\/jats:italic>.<\/jats:p><jats:p>As G. M. Bergman has pointed out (in a letter to the author), it is necessary to specify rather more precisely what type of problem we assume to be soluble for <jats:italic>K<\/jats:italic>: We must be able to decide whether or not a given finite set in <jats:italic>K<\/jats:italic> is linearly dependent over its centre. This makes it desirable to prove that the free field has a corresponding property (and not merely a soluble word problem). This is done in \u00a72; interestingly enough it depends only on the solubility of the word problem in the free field (cf. Lemma 2 and Theorem 1\u2032 below).<\/jats:p><jats:p>Bergman also notes that the proof given in [1] does not apply when <jats:italic>K<\/jats:italic> is finite-dimensional over its centre; this oversight is rectified in \u00a74, while \u00a73 lifts the restriction on <jats:italic>C<\/jats:italic> (to be infinite). However, we have to assume <jats:italic>C<\/jats:italic> to be the precise centre of <jats:italic>K<\/jats:italic>, and not merely a central subfield, as claimed in [1].<\/jats:p><jats:p>I am grateful to G. M. Bergman for pointing out the various inaccuracies as well as suggesting remedies.<\/jats:p>","DOI":"10.2307\/2272273","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:33:55Z","timestamp":1146936835000},"page":"69-74","source":"Crossref","is-referenced-by-count":8,"title":["The word problem for free fields: a correction and an addendum"],"prefix":"10.1017","volume":"40","author":[{"given":"P. M.","family":"Cohn","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200054293_ref001","first-page":"309","volume":"38","author":"Cohn","year":"1973","journal-title":"The word problem for free fields"},{"key":"S0022481200054293_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-12-291350-1.50010-3"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200054293","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T16:06:16Z","timestamp":1559145976000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200054293\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,3]]},"references-count":2,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1975,3]]}},"alternative-id":["S0022481200054293"],"URL":"https:\/\/doi.org\/10.2307\/2272273","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,3]]}}}