Studies in Rings generalised Unique Factorisation Rings

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-31- Right bounded HNP rings are HNP rings in which every essential right ideal contains a two sided ideal. Le na qa n [9] has shown tha t right bounded HNP rings are rings with er.ouqh invertible ideals, ioe, in such rings every nonzero prime ideal contains invertible ideals o Thus~ a second way to look at prime GUFRs is through their connection with prime Noetherian rings with enough invertible ideals. It can be seen that if a prime Noetherian ring R with enough invertible ideals is such that all its invertible ideals are principal, then R is a prime GUFR. In particular, right bounded HNP rings in which each invertible ideal is principal, are also prime GUFRs. After proving all the above mentioned results in Chapter 2, we move over to Chapter 3 in ~hich we study different extension rings of GUFRs. A finite central extension [10 (pp. 343-77)J ring S of a GUFR R is shown to be a GUFR if the regular elements of R are also regular elements in S. As a consequence the n x n matrix ring Mn(R) over any GUFR,R, is f ound to be a GUFR. R] x , Cl], the ring of polynomials, twisted by an automorphismJover a GUFR [11J and R[x,SJ, the ring of polynomials/twisted by a derivation $ over a GUFR [12J are investigatedo
-32- The concept of a ring with few zero divisors [13] in the commutative case is generalised to the non-commutative case and the idea of weakly invertible elements is introduced. Some analogous results of quasi-valuation rings [13] in the non-commutative case have been proved. We conclude Chapter 3 with a discussion of integral closure [14J, [15J of a GUFR. The technique of localisation at a Commutative Noetherian rings, cannot be prime ideal in brought into non commutative Noetherian rings as it is. This is because of the gener~l behaviour of prime ideals in non-Commutative rings. This is a major problem (ioe., under what conditions, can a Noetherian ring be localised at its prime ideals?) still confronting the study of non-Commutative Noetherian rings. At present, a theory has emerged as the correct one. Jategaonkar [16], Muller [17] etco are some of the forerunners in this study. We give a detailed disc~sion of this recent development in the localisation at prime ideals in Noetherian rings in Chapter 4 and identify some prime ideals and cliques of prime ideals at which the localisation is possible in GUFRs. In chapter 5, we discuss some problems that arose in the thesis whi~h are to be investigated. Also, n possible extension of the concept of GUFR to non-Noetherian case is given. The preliminary materials of this chapter have been taken from [18J, [19J, [20J and [21J.
Page 1 and 2: STUDIES IN RINGS GENERALISED UNIQUE
Page 3 and 4: \CKN0 1vVLEDGEWIENT Chapter 1 INTRO
Page 5 and 6: -2- REMIl'JDER OF Tt-iE COM~1UTAT l
Page 7 and 8: -4- non zero prime ideal of R conta
Page 9 and 10: -6- If both conditions hold, R is s
Page 11 and 12: -8- In non-commutative rings, it tu
Page 13 and 14: -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: -30- It is found that the elements
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 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 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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