Studies in Rings generalised Unique Factorisation Rings

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-109- Jk < Qo Consequently J ~ Q, since Q is prime. Thus I ~ Q, which contradicts the selection of Q. Therefore P satisfies the right second layer condition. Corollary 4.36. Let P be a height 1 prime ideal of a GUFR such that {p} is right stable. Then P is classically right localisableo Proof: This is an immediate c ons equen c e of theorem 4.14 and theorem 4035. A semiprime ideal S of a Noetherian ring is said to be classically right localisable if the finite set of prime ideals associated with S is classically right localisable. Thus we get another consequence of theorenl 4.35 and theorem 4.23. Corollary 4.37. Let S be a semiprime ideal in a GUFR and assume that the associated prime ideals of S are height 1 prime ideals. Suppose also that the collection of associated primes is right stable. Then S is classically r i.qh t Lo c a.l i s a bl e ,
-1 lC)- Defini.tion 4 038. Let R be a Noetherian ring. A subset X of Spec R is said to be a .§J2arse subset if, given any Q € .Spec R and given any c £ CR(Q), we have Remark 4 039. Let R be a GUFR and I be a normal invertible ideal of R. Put X I = '.p, Spec RI height P = 1 and I < pJ Then by principal ideal theorem XI /: (fjo LetQ ~ Spec 11 and c £ C(Q). Then, if Q is minimal,! cannot be contained in Q, since R is a GUFR, whereas n{p ~ XII Q ins I. Further, if Q is nonminimal, then height of Q ~ 1 and so there exists no height 1 prime P such that Q in both cases o f:: n{p f xr/o< P, c f- CR(P)J. Therefore X r is a s pa r s e set in R. The 0 r ern 4 0 40 • Let R be a GUFR and I be a normal invertible ideal of R. Also assume that for a prime idealQ,{PE. XI/Q ~pJ is right s t ab Le , Then [p E. Xr/O ~pJ is a c La s s i c a l Ly right localis~ble seto
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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
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 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: -108- M annihilated by I, I~(lV~) =
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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