Studies in Rings generalised Unique Factorisation Rings

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-41- Corollary 2 08. If R is an NUFR, then T is a simple ring. Artinian rings are generally regarded as generalisa~ tion of s emi s i.rnp I e Artinian rings. Goldie's theorem gives a cha ra c terisa tion 0 f thos e ring 5 wh Let: a re orde rs in semisimple Artinian rings. This result naturally gives rise to the question: 'tJhich rings can be orders in Artinian "rings? The importance of Artinian quotient rings is that they will be useful in the study of localisation at a prime ideal in Noetherian rings and in the study of finitely generated torsion free modules over Noetherian r i nq s , It is seen that there are Noetherian ri.ngs whi ch lack Art i n i an quotient rings. However, if R is a GUFR, R always have an Artinian quotient ring. We prove this next. Every GUFR has an Artinian quotient ringo Proof From the definition of a GUFR, every non-minimal prime idea1 c ont a ins no rma 1 i nvertib1 e idea 1 s • Th e 9 enerators 0 f these norma 1 anve r t i b I e idea 1 s a re in eR (0) (the 5 e t 0 f regular elements of R), by theorem 2 0 6 . Now the theorem follows from proposition 1.63, which states that R is a
-42- Noetherian order in an Artinian ring if and only if {PE Spee Rip n CR(O) =~] $ Min Spee R. Remark 2 010. Thus, even though in the definition of GUFR, we are not a s scn.inq that it has an Artinian quotient ring, it turns out that GUFRs always have Artinian quotient rings. It is also obvious that the Artinian quotient -ring is an ov~r-ring of the GUFR, and the so called S-invertible ideals are invertible with respect to this Artinian quotient ring also. Hence the terminologies, over-ring Sand S-invertible ideals, can be avo i d od in the definition of" a GUFR. Definition 2.11. Let X be a right denominator set in a ring Ro If I is a right ideal of RX- l, the set {a E RI aI-lE I} is called th~ contraction of I to R and 'r;;:~ is denoted by rC. -.6 If J is a right ideal of R, then lex-lie E I,x E x} in RX- 1 is called the extension of J in RX- 1 and is denoted by Je . Proposition 2.12. Let X be a right denominator set in a right Noetherian ringo Then
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 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: -40- P.O C = ~ for i = 1 , 2 , ...
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
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-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
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-io>- Theorem 4 023. If R is a Noet
Page 107 and 108:
Lemma 4.28. Suppose P and Q are max
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-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
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Chapter-5 REMARKS In this concludin
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-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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