Studies in Rings generalised Unique Factorisation Rings

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-63- Theorem 2.44c If R is a right bounded hereditary Noetherian prime ring in which every invertible ideal is' normal. Then R is a prime GUFR. Proo f: Since every invertible ideal is normal, the theorem follows from lemma 2 0 38 0 COMPLETELY FAITHFUL MODULES Definition 2.45. Let R be a ring and M be a right R-module. Then l\~ is (1) Unfaithful if it is not faithful. (2) Completely faithful if A/B is faithful for all submodules A ) B of M. (3) Local~y unf~ithful provided every finitely generated submodule is unfaithful. (4) Locally Artinian provided every finitely generated submodule is Artinian. From theorem 2 040 and the results [23, lemma 2 0 4 , theorem 2.6],we get the following theorems for a prime GUFR.
-64- Theoren12.46. Let R be a pri.me GUFR' and lv\ be a cyclic R-rnodule 0 If N is a submodule of M such that (1) N is completely faithful and M/N unfaithful, or (2) N is unfaithful and M/N completely faithful Q Then N is a direct summand of Mo Proof: As in [ 24, lemma 2.3J. Theorem 2.47. Let R be a prime GUFR and A 98,C right R-moduleso Then t.h e exact sequence 0 ----t A ---+ B ~ C -----.,. 0 splits provid9d anyone of the following statements holds. (1) A is completely faithful and C is locally unfa.i t.h f u L, (2) A is unfaithful and C is completely faithful, (3) A is locally unfaithful and C is completely faithfulo Proof: As in [24, theorem 2 04J. Remark 2 04.8. For any macule M, it can be proved, using Zornvs
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STUDIES IN RINGS GENERALISED UNIQUE
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\CKN0 1vVLEDGEWIENT Chapter 1 INTRO
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-2- REMIl'JDER OF Tt-iE COM~1UTAT l
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-4- non zero prime ideal of R conta
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-6- If both conditions hold, R is s
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-8- In non-commutative rings, it tu
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-10- The next two propositions guar
Page 15 and 16: -12- (a) I is a semiprime ideal (b)
Page 17 and 18: -14- Definition 1 029. An ideal P i
Page 19 and 20: -16- Proposition 1 033. 'equiva len
Page 21 and 22: -18- Proposition 1038. For a ring R
Page 23 and 24: -20- there exists d ' E:. 0 such th
Page 25 and 26: -22- expression. Then it can be int
Page 27 and 28: -24- Definition 1 0 4 9 . A regul~r
Page 29 and 30: -26- that the right ideal generated
Page 31 and 32: -28- Proposition 1.62. Let R be a r
Page 33 and 34: -30- It is found that the elements
Page 35 and 36: -32- The concept of a ring with few
Page 37 and 38: -34- universal enveloping algebra-
Page 39 and 40: -36- (3) We give an example of a GU
Page 41 and 42: -38- Let R be a GUFR with over-ring
Page 43 and 44: -40- P.O C = ~ for i = 1 , 2 , ...
Page 45 and 46: -42- Noetherian order in an Artinia
Page 47 and 48: -44- Proof Since R is a Noetherian
Page 49 and 50: -46- In the proof of le~ma 2.17 we
Page 51 and 52: -48-- ( 1) CR(P) 5 CR(O) for each h
Page 53 and 54: -50- Remark 2.22. Prime GUFRs are t
Page 55 and 56: -52- Proof: As in [22, Proposition
Page 57 and 58: -54- Theorem 2.28. Let R be a Noeth
Page 59 and 60: -56- Lemma 2 032. Let R be a prirne
Page 61 and 62: Therefore, in both cases we have pr
Page 63 and 64: -60- RI~JGS yVITli [::NOUGH INVER~r
Page 65: -62- Th e o r ern 2.43. Let R be a
Page 69 and 70: Chapter-3 EXTENSIONS AND RINGS WITH
Page 71 and 72: -68- If S is a finite normalising e
Page 73 and 74: -70- regular element of R is regula
Page 75 and 76: -72- Lemma 3.110 Let P be a prime i
Page 77 and 78: -74- Now, if x ( P, then xS = Sx is
Page 79 and 80: -76- so that Sg = gS $ P. Since S i
Page 81 and 82: -78- is an r' E R such that Equa t
Page 83 and 84: -80- As in [12, lemma 1 03]. Lemma
Page 85 and 86: -82- Examples 3023 (1) In [13J, a c
Page 87 and 88: -84- each j which is not possible.
Page 89 and 90: -86- Thus the ideal T(z+ux)T is not
Page 91 and 92: -88- in the generating set of I suc
Page 93 and 94: -90- Proof First we prove that R is
Page 95 and 96: Chapter 4 INTRODUCTION LOCALISATION
Page 97 and 98: -94- Now i f [~~] ~ C( Q) a nd [~~
Page 99 and 100: -9()- Remark 4 05. It can be seen t
Page 101 and 102: -98- Remark 4 012. Let P be a class
Page 103 and 104: -100- Lemma 4 019. An invertible id
Page 105 and 106: -io>- Theorem 4 023. If R is a Noet
Page 107 and 108: Lemma 4.28. Suppose P and Q are max
Page 109 and 110: -106- and P is primeo Cons equen t
Page 111 and 112: -108- M annihilated by I, I~(lV~) =
Page 113 and 114: -1 lC)- Defini.tion 4 038. Let R be
Page 115 and 116: Chapter-5 REMARKS In this concludin
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-114- -1 -1 and so a I = R and a a
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REFERENCES [1] AoW. Chatters (1984)
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-118- [17J B.J. Muller Localisation
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Studies in Rings generalised Unique Factorisation Rings

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