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137:, it follows that a local, right hereditary ring is a right fir, and a local, right semihereditary ring is a right semifir.
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Since all right ideals of a right fir are free, they are projective. So, any right fir is a right
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is a right fir, since every nonzero principal right ideal of a domain is isomorphic to
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260:, London Mathematical Society Monographs, vol. 19 (2nd ed.), Boston, MA:
68:≥ 0.) The semifir property is left-right symmetric, but the fir property is not.
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right ideals are free modules of unique rank. (Thus, a ring is semifir if it is
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92:. These last facts are not generally true for noncommutative rings, however (
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Unlike a principal right ideal domain, a right fir is not necessarily right
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since it is a hereditary domain, and so is necessarily
Noetherian.
290:, Springer Undergraduate Mathematics Series, Berlin, New York:
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generators are free and have unique rank is called an
200:, vol. 1, Gauthier-Villars, pp. 273–278,
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228:Free ideal rings and localization in general rings
44:. A ring such that all right ideals with at most
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348:Skew fields. Theory of general division rings
194:"Free ideal rings and free products of rings"
88:, while a commutative semifir is precisely a
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230:, New Mathematical Monographs, vol. 3,
76:It turns out that a left and right fir is a
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133:are free, and because local rings have
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165:non-commutative polynomial rings
258:Free rings and their relations
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159:-algebras for division rings
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110:. In the same way, a right
101:principal right ideal domain
322:Encyclopedia of Mathematics
288:Introduction to ring theory
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380:
354:Cambridge University Press
232:Cambridge University Press
32:, is a ring in which all
178:and every semifir is a
72:Properties and examples
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453:Abstract algebra stubs
397:-related article is a
176:invariant basis number
135:invariant basis number
86:principal ideal domain
286:Cohn, P. M. (2000),
256:Cohn, P. M. (1985),
192:Cohn, P. M. (1971),
123:semihereditary ring
84:fir is precisely a
127:projective modules
80:. Furthermore, a
58:finitely generated
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317:"Free ideal ring"
301:978-1-85233-206-8
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224:Cohn, P. M.
131:local rings
125:. Because
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22:ring theory
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442:Categories
372:0840.16001
342:Cohn, P.M.
216:2010-11-26
186:References
142:Noetherian
327:EMS Press
171:, §5.4).
169:Cohn 2000
94:Cohn 1971
344:(1995),
226:(2006),
329:, 2001
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