Studies in Rings generalised Unique Factorisation Rings

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-65- lemma, that there existr a unique maximal completely faithful submodule C(M), which contains every completely faithful submodule of M. [24, lemma 2 0 2J. Theorem 2 049. Let R be a prime GUFR and let M be a locally Artinian right R-module. Then there exists a locally unfaithful submodule N of M such that M = C(M) e N. Proof As in [24, theorem 2.6J.
Chapter-3 EXTENSIONS AND RINGS WITH MANY NORMAL ELEMENTS , INTRODUCT ION In this chapter, we discuss rings which are extensions of GUFRs namely the finite centralising extensions, Ore extension and the ring of polynomials twisted by a derivation. Also we introduce the concept of rin~s with many normal elements. We show that any finite centralising e xt.e n s i o n of a GUFR is a GUFR 0 As a corollary of this result, M (R), the n n x n matrix ring, over a GUFR R is a GUFR. A sufficient condition for the Ore extension, over a Noetherian ring with Artinian quotient ring, to be a GUFR is obtained. The Noetherian rings with Artinian quotient rings such that the Ore extensions over them are prime GUFRs a r e characterised. The skew polynomial rings over some special Noetherian rings are investigatedo We extend the concept of rings with few zero divisors[13] in the commutative case to rings with many normal elements in the non-ecomrnu t a t.Ive case. By introducing the concept of weakly invertible elements, we study some properties of Noetherian rings with many normal elements. Also, we prove some results.on the integral closure of Noetherian rings, in this chapter.
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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
Page 9 and 10:
-6- If both conditions hold, R is s
Page 11 and 12:
-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)
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 and 34: -30- It is found that the elements
Page 35 and 36: -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: -64- Theoren12.46. Let R be a pri.m
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
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REFERENCES [1] AoW. Chatters (1984)
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-118- [17J B.J. Muller Localisation
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