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We also describe a way to construct weak standard bases and reduced standard bases of ideals of a free associative algebra over an arbitrary finitely generated ring (over a finitely generated algebra over a field). Some examples of constructions of standard bases and of solutions of the equality problem are included. <\/jats:p>","DOI":"10.1142\/s021819679800034x","type":"journal-article","created":{"date-parts":[[2003,7,31]],"date-time":"2003-07-31T10:08:41Z","timestamp":1059646121000},"page":"689-726","source":"Crossref","is-referenced-by-count":19,"title":["Standard Gr\u00f6bner-Shirshov Bases of Free Algebras Over Rings, I"],"prefix":"10.1142","volume":"08","author":[{"given":"Alexander A.","family":"Mikhalev","sequence":"first","affiliation":[{"name":"Department of Mechanics and Mathematics, Moscow State University, Moscow, 119899, Russia"}]},{"given":"Andrej A.","family":"Zolotykh","sequence":"additional","affiliation":[{"name":"Department of Mechanics and Mathematics, Moscow State University, Moscow, 119899, Russia"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"p_2","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1986-0846601-5"},{"key":"p_3","doi-asserted-by":"publisher","DOI":"10.1016\/S0747-7171(88)80053-1"},{"key":"p_4","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-51517-8_156"},{"key":"p_5","doi-asserted-by":"publisher","DOI":"10.1016\/0022-314X(83)90021-5"},{"key":"p_6","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0016860"},{"key":"p_10","first-page":"198","author":"Bayer D.","year":"1993","journal-title":"Sympos. 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