Studies in Rings generalised Unique Factorisation Rings

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-27- Goldie's theorems give the structure of a semiprime (prime) right Goldie ring, as it is a right order in a semisimple Artinian (simple Artinian) ring which in turn is the finite direct product of matrix rings over division rings (matrix ring over a division ring). Thus, in particular we get the structure theorems for semiprime (prime) right Noetherian rings. ARTINIAN QUOTIENT RINGS In the previous section we have seen that every right Noetherian semiprime ring (every right Noetherian prime ring) is a right order in a semisimple (simple) Artinian ring. Now we see the more general case, i.eo, when Q, the quotient ring, is simply an Artinian ringo Some times we call Q the total quotient ring as it consists of all quotients with denominators varying over the regular elements. Proposition 1 061. Let R be a ring which has a right quotient ring Q which is right Artinian and let A be an ideal of R, then AQ is an ideal of Q.
-28- Proposition 1.62. Let R be a right Noetherian ring with N, the prime radical and of R. Then, Pl,P2, ••. ,Pn the minimal prime ideals (1) The right regular elements are regular modulo N (3) Let a,e ~ R with c right r-equI a r , then there exists be::: Rand dt::;CR(N) such that ad=bc. (4) R has a right Artinian quotient ring if and only if, CR(O) = CR(N). "\Ne state a result, a characterisation of Noetherian rings which are orders in Artinian rings, proved by PoF o Smith [4]. Proposition 1 0 63 . A Noetherian ring R is an order in an Artinian ring if and only if t: (;- Spec Rip n C R (0) =~} ~ Minimal (Spec R) where Spec R denotes the collection of all prime ideals of R.
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: -26- that the right ideal generated
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 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
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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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