Studies in Rings generalised Unique Factorisation Rings

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-~1-~tA7- -99- Definition 4 015. An ideal I of R is said to have the right Artin-Rees (Arrl-2£Qper~y if for any finitely generated right R module M containing an essential submodule L with LI = 0, there is a positive integer n such that n MI = O. In this case we call I a right AR Ldea l , Left AR property is defined analogously. Re ma r k Lt 4 1 () . ( 1) A prime idea 1 P wi th the ri gh t AR property always satisfies the right second layer conditiono ( 2 ) Ani(j E: a 1 I 0 f R i srigh tAR i fandon 1y i f for every right ideal K of R, there is a positive integer n such that KC\I n < KI. Theorem 4 017. If R is a Noetherian ring and P is a prime ideal with the right AR property, then P is classically localisable if and only if there is no prime ideal Q of R with P < Q and Q ~ P. Lemma 4.18. Suppose an ideal in a right Noetherian ring R has the right AR p rop e r t v , If Q ~ P in Spec R and if I .$ P, then I S Q.
-100- Lemma 4 019. An invertible ideal in a (right) Noetherian ring has the (right) AR property. Let R be a Noetherian ring with a quotient ring Q. , J Let P = aR = ha be a prime ideal with a regularo Then R has a partial quotient ring S obtained by localising R at the Ore set {l,a,a 2 •.• J It is easy to see that P = aR = Ra is S-invertible and hence P has the (right and left) AR property. Using induction and regularity of a, it is easy to see that C(p) ~ C(pn) for every n. Now by [28, proposition 2 0 1 ] , P is localisable which gives the proof of lemma 2.17. Given a prime ideal P, any right localisation at P must be found by inverting a right Ore set C ~ C R (p) · Thus, in fact, C ~ () [CR(Q) IQ , rt cl p} by corollary 4 07. Let X != Spec R and define C(X) = () { C ( Q ) R IQ' XJ. If X is a right clique and if we want to localise at X, then c(x) must be a right Ore set. We also want some nice properties for the quotient ring. Definition 4 0 20 . Let R be a Noetherian ring and X < Spec R. Then
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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
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 and 66: -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: -98- Remark 4 012. Let P be a class
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 and 118: -114- -1 -1 and so a I = R and a a
Page 119 and 120: REFERENCES [1] AoW. Chatters (1984)
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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