387:
428:
299:
269:
239:
137:, it follows that a local, right hereditary ring is a right fir, and a local, right semihereditary ring is a right semifir.
421:
361:
452:
193:
164:
414:
326:
321:
353:
231:
316:
57:
117:
Since all right ideals of a right fir are free, they are projective. So, any right fir is a right
155:
Another important and motivating example of a free ideal ring are the free associative (unital)
175:
134:
100:
85:
41:
447:
309:
279:
249:
205:
77:
371:
8:
402:
122:
33:
346:
106:
is a right fir, since every nonzero principal right ideal of a domain is isomorphic to
357:
295:
265:
235:
126:
394:
367:
260:, London Mathematical Society Monographs, vol. 19 (2nd ed.), Boston, MA:
179:
81:
111:
89:
305:
291:
275:
245:
201:
149:
141:
118:
68:≥ 0.) The semifir property is left-right symmetric, but the fir property is not.
60:
right ideals are free modules of unique rank. (Thus, a ring is semifir if it is
398:
261:
92:. These last facts are not generally true for noncommutative rings, however (
441:
352:, Encyclopedia of Mathematics and Its Applications, vol. 57, Cambridge:
140:
Unlike a principal right ideal domain, a right fir is not necessarily right
37:
21:
17:
130:
209:
341:
223:
386:
152:
since it is a hereditary domain, and so is necessarily
Noetherian.
53:
290:, Springer Undergraduate Mathematics Series, Berlin, New York:
198:
48:
generators are free and have unique rank is called an
200:, vol. 1, Gauthier-Villars, pp. 273–278,
345:
228:Free ideal rings and localization in general rings
44:. A ring such that all right ideals with at most
439:
422:
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
230:, New Mathematical Monographs, vol. 3,
76:It turns out that a left and right fir is a
429:
415:
121:, and likewise a right semifir is a right
71:
133:are free, and because local rings have
440:
381:
340:
285:
255:
222:
191:
168:
93:
144:, however in the commutative case,
13:
334:
14:
464:
385:
165:non-commutative polynomial rings
258:Free rings and their relations
1:
185:
159:-algebras for division rings
20:, especially in the field of
401:. You can help Knowledge by
110:. In the same way, a right
101:principal right ideal domain
7:
322:Encyclopedia of Mathematics
288:Introduction to ring theory
10:
469:
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
56:is a ring in which all
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
410:
409:
317:"Free ideal ring"
301:978-1-85233-206-8
271:978-0-12-179152-0
241:978-0-521-85337-8
460:
431:
424:
417:
395:abstract algebra
389:
382:
374:
351:
330:
312:
282:
252:
219:
218:
217:
208:, archived from
180:Sylvester domain
468:
467:
463:
462:
461:
459:
458:
457:
438:
437:
436:
435:
378:
364:
337:
335:Further reading
315:
302:
292:Springer-Verlag
272:
242:
215:
213:
188:
150:Dedekind domain
119:hereditary ring
74:
26:free ideal ring
12:
11:
5:
466:
456:
455:
450:
434:
433:
426:
419:
411:
408:
407:
390:
376:
375:
362:
336:
333:
332:
331:
313:
300:
283:
270:
262:Academic Press
253:
240:
220:
187:
184:
174:Semifirs have
163:, also called
114:is a semifir.
73:
70:
9:
6:
4:
3:
2:
465:
454:
451:
449:
446:
445:
443:
432:
427:
425:
420:
418:
413:
412:
406:
404:
400:
396:
391:
388:
384:
383:
379:
373:
369:
365:
363:0-521-43217-0
359:
355:
350:
349:
343:
339:
338:
328:
324:
323:
318:
314:
311:
307:
303:
297:
293:
289:
284:
281:
277:
273:
267:
263:
259:
254:
251:
247:
243:
237:
233:
229:
225:
221:
212:on 2017-11-25
211:
207:
203:
199:
195:
190:
189:
183:
181:
177:
172:
170:
166:
162:
158:
153:
151:
147:
143:
138:
136:
132:
128:
124:
120:
115:
113:
112:BĂ©zout domain
109:
105:
102:
97:
95:
91:
90:BĂ©zout domain
87:
83:
79:
69:
67:
64:-fir for all
63:
59:
55:
51:
47:
43:
39:
35:
31:
27:
23:
19:
403:expanding it
392:
377:
347:
320:
287:
257:
227:
214:, retrieved
210:the original
197:
173:
160:
156:
154:
145:
139:
116:
107:
103:
98:
75:
65:
61:
49:
45:
40:with unique
38:free modules
34:right ideals
29:
25:
24:, a (right)
15:
448:Ring theory
224:Cohn, P. M.
131:local rings
125:. Because
82:commutative
22:ring theory
18:mathematics
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
310:1732101
280:0800091
250:2246388
206:0506389
54:semifir
370:
360:
308:
298:
278:
268:
248:
238:
204:
99:Every
78:domain
393:This
148:is a
129:over
50:n-fir
28:, or
399:stub
358:ISBN
296:ISBN
266:ISBN
236:ISBN
52:. A
42:rank
36:are
368:Zbl
96:).
30:fir
16:In
444::
366:,
356:,
325:,
319:,
306:MR
304:,
294:,
276:MR
274:,
264:,
246:MR
244:,
234:,
202:MR
196:,
182:.
430:e
423:t
416:v
405:.
167:(
161:k
157:k
146:R
108:R
104:R
66:n
62:n
46:n
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.