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193:. In general, a module is simple if and only if it is nonzero and is generated by each of its nonzero elements.
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is considered as a left module over itself, then its cyclic submodules are exactly its left
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that is generated by one element. The concept is a generalization of the notion of a
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423:, vol. 13 (2 ed.), New York: Springer-Verlag, pp. x+376,
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490:(Third ed.), Reading, Mass.: Addison-Wesley, pp. 147–149,
288:; there may also be other cyclic submodules with different
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269:are cyclic submodules. (The Jordan blocks are all
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314:, there exists a canonical isomorphism between
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403:
415:Anderson, Frank W.; Fuller, Kent R. (1992),
48:-module) that is generated by one element.
72:can be generated by a single element i.e.
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386:
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181:generated by any non-zero element
14:
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460:Rings, modules and linear algebra
189:is necessarily the whole module
417:Rings and categories of modules
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380:
204:as a ring. The same holds for
1:
464:. Chapman and Hall. pp.
421:Graduate Texts in Mathematics
406:, Just after Proposition 2.7.
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177:is a cyclic module since the
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7:
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152:-module is a cyclic module.
138:
10:
538:
404:Anderson & Fuller 1992
429:10.1007/978-1-4612-4418-9
368:Finitely generated module
246:-module which is also a
390:Algebra I: Chapters 1–3
20:, more specifically in
456:; T.O. Hawkes (1970).
310:that is generated by
107:. Similarly, a right
229:ring of polynomials
248:finite-dimensional
34:module over a ring
497:978-0-201-55540-0
30:monogenous module
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214:mutatis mutandis
202:principal ideals
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302:Given a cyclic
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292:; see below.)
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115:is cyclic if
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42:Abelian group
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26:cyclic module
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350:
345:denotes the
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290:annihilators
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251:vector space
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186:
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38:cyclic group
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484:Lang, Serge
347:annihilator
257:, then the
208:as a right
22:ring theory
18:mathematics
506:0848.13001
454:B. Hartley
387:Bourbaki,
374:References
297:Properties
271:isomorphic
265:acting on
64:is called
52:Definition
212:-module,
179:submodule
125:for some
99:for some
516:Category
486:(1993),
362:See also
333:, where
306:-module
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139:Examples
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488:Algebra
468:, 152.
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