Studies in Rings generalised Unique Factorisation Rings

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-115- In chapter 4, theorem 4.35 assures the right second layer condition for height one prime ideals in a GUFR. The right second layer condition for a prime ideal of height> 1 in a GUFR is yet to be discussed. Also, it is not yet investigated whether {p €. Xr/Qr--t p} in theorem 4.4J, is always right stable or not. However, from [26], it follows that, in a GUFR if every height 1 prime ideal is maximal, then each XI is right stable, satisfies the right second layer condition (theorem 4.35) and the incomparability condition. It may be possible to extend the concept of GUFRs to (non-Noetherian) rings with (left and right) Krull dimension [30]. The analogous nature of such rings with Noetherian rings is a major source of interest in them. The invertible ideals, in rings with (left and right) Krull dimension, also behave well o A study of invertible ideals in rings with Krull dimension is given in [31].
REFERENCES [1] AoW. Chatters (1984) Non-commutative unique factorisation domains, Math.Proc.Cambridge Phi10. Soc., 95,49-54. [2J A.W. Chatters and D.A. Jordan (1986) Non-commutative unique factorisation rings, JoLondon Math.Soc.(2)33, 22-32. [3] I. Kaplanski (1974) Commut a t i ve Rings, Chicago Press. [ 4 ] P • F. Smith (1972) On two sided Artinian quotient rings, Glasgow Math. 3.(13), 159-163. [5] A.W. Chatters (1972) A decomposition theorem for Noetherian hereditary rings, Bull.London Math.Soc. 4, 125-12.6. [6] A.W. Chatters and C.R. Hajarnawis (1980) Rings with chain conditions, Pitman, London. [7J C. Faith (1976) Algebra 11, Ring Theory, Springer Verlag.
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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
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-12- (a) I is a semiprime ideal (b)
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-14- Definition 1 029. An ideal P i
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-16- Proposition 1 033. 'equiva len
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-18- Proposition 1038. For a ring R
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-20- there exists d ' E:. 0 such th
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-22- expression. Then it can be int
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-24- Definition 1 0 4 9 . A regul~r
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-26- that the right ideal generated
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-28- Proposition 1.62. Let R be a r
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-30- It is found that the elements
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-32- The concept of a ring with few
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-34- universal enveloping algebra-
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-36- (3) We give an example of a GU
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-38- Let R be a GUFR with over-ring
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-40- P.O C = ~ for i = 1 , 2 , ...
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-42- Noetherian order in an Artinia
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-44- Proof Since R is a Noetherian
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-46- In the proof of le~ma 2.17 we
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-48-- ( 1) CR(P) 5 CR(O) for each h
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-50- Remark 2.22. Prime GUFRs are t
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-52- Proof: As in [22, Proposition
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-54- Theorem 2.28. Let R be a Noeth
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-56- Lemma 2 032. Let R be a prirne
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Therefore, in both cases we have pr
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-60- RI~JGS yVITli [::NOUGH INVER~r
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-62- Th e o r ern 2.43. Let R be a
Page 67 and 68: -64- Theoren12.46. Let R be a pri.m
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
Page 117: -114- -1 -1 and so a I = R and a a
Page 121 and 122: -118- [17J B.J. Muller Localisation
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Studies in Rings generalised Unique Factorisation Rings

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