Studies in Rings generalised Unique Factorisation Rings

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-85- Theorem 3.29. Let R be a Noetherian ring with many normal eleme nts and T be the partial quo tiffi t ring of R at C =(a € R/a is normal) . Then for any z in T and x c: C, there is ~, an element u such that z + ux is weakly T-invertible. Proof: By remark 3.~4, T is an over-ring of Rand T has only a finite number of maximal idealso Let ~ = [M/M is a maximal ideal of T with z e M] and ~. = (M/M is a maximal ideal of T with z ~ M] Then ..0 and ill are finite collections of prime ideals and nei the r M S. M I nor M' ~ M for any 1\1 E ~ and ~~lEd' as they are ma~imal idealso Thus by theorem 3.26, there exists an element u 6: M, for all M E AI and u tf. M for any M€ A • Then the eleme nt z+ux ~ M for any maximal ideal of To For, if z+ux c M for some M€Ll, then z+ux-z ux .. t:; = M. But x
-86- Thus the ideal T(z+ux)T is not contained in any maximal ideal of T and hence (z+ux) is weakly I-invertible. Theorem 3 0 30 . Let R be a Noetherian ring with many normal elements and I be a one sided ideal of R containing a normal element. Then I can be generated by a set of weakly T-invertible elements. Proof: Suppose I is a left ideal. Let (Zl'Z2 p • • ,zJ be ageneratin9 set 0 f I a nd x be anar ma 1 e 1 erne ntin I. Consider zl' by theorem 3.29, there exists an element U 1 , such tha .~ zl+u1'x is weakly T-invertibleo Since I -1 u = u1c for "i € R 1 some anti Cl € e, we have .l -1 u 1 = u1l = not containing U 1 u1'c l does not belong to the maxima 1 ideals I and u 1 belongs to all maximal ideals, which contains u 1 ' . Thus as in the proof of theorem 3 0 2 9 , Zl + u1x is a weakly I-invertible element in R. Similarly we get a collection (Zi+Uix] of weakly T-invertible elements for each zi. Since x is normal, x £ C and so x is invertible in T. Thus [zi+uix,x} is a collection of weakly T-invertible elements in 10 We prove this is generating set for 10
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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
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 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: -84- each j which is not possible.
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 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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