Studies in Rings generalised Unique Factorisation Rings

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-81- Le t E = fb f R / bR = Rb is an invertible ,f -ideal] • Then E is an Ore set consisting regular elements and the localised ring T, of R at E, is an ~-simple ring, ioeo, o and T are the only n-ideals of T, where ~ is the extension of J to T defined by ~(ac -1) = J(a)c- l for -1 all ac ~ T. If P is a non-minimal prime ideal of S, then as in the pattern of the proof of the theorem 3 013, we have, PT[x, cl ] = fT[x, d ] = rt». et ]f J for some f € T[x,J] and we ge t f = gd- l for some 9 L:' ~ P an d d r: ~ E and we h aye gR = Rg. Also, xg = xfd = fxd = f(dx + J(d)) = fdx+£i(d) = fdx + fdu, for some u£ R. Thus xg = fd(x+u) = g(x+u) and it follows that 5g 5 g5 and similarly 95 ~ Sg. Thus gS = 5g is contained in P and is Q(S)-invertible, since 9 E CS(O). RINGS WI1~1-1 ~NY NORN\AL ELEMENTS In thi~ section we introduce the concept of many normal elements, which is a generalisation of GUFRs o By a normal element, we mean a normal regular element in this sectiono Definition 3.22. Let R be any ringo Then R is called a ring with many normal elements if R has only a finite number of prime ideals not containing any normal elements.
-82- Examples 3023 (1) In [13J, a commutative ring with few zero divisors are defined as any commutative ring with only a finite number of maximal O-ideals, where a maximal O-ideal is an ideal maximal with respect to not containing non-zero divisors. Since every maximal O-ideal is a prime ideal, it follows that in the commutative case, rings with few zero-divisors are rings with many normal elements. (2) If R is a GUFR, then every non-minimal prime ideal contains a normal element. Since the number of minimal prime ideals in any Noetherian ring is finite, it follows that every GUFR is a ring with many normal elements. Remark 3.24. Let R be a Noetherian ring with many normal elements and C = {a (R/aR = Ra is normal} • Then as in theorem 2.7, it can be prov e c; that C is an Ore set and the localised ring T = RC- l = C-1R has only a finite number of maximal ideals, precisely the extensions of the prime ideals of R not containing normal elements. Also T is an over-ring, since C has only regular elementso Vve s tare a theorem known as t h e "prime avoidance" theorem [20, proposition 2012 0 7 ] .
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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
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 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: -80- As in [12, lemma 1 03]. Lemma
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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