Studies in Rings generalised Unique Factorisation Rings

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-17- Artinian where B is a submodule of the module A and that any finite direct sum of Artinian modules is Artinian. Also, if R is an Artinian ring, so is every finitely 'generated R-module. Now we state the celebrated theorems on simple and semisimple rings, due to Weddernburn and Artin. Proposition 1 037. For a ring R, the following conditions are equivalent. (a) (b) (c) (d) R is right Artinian and J(R) = 0 R is left Artinian and J(R) = 0 R is semisimple R = M (0 1) x M (02) x ... x M (D k) for some n 1 n 2 n k positive.integers n 1,n2, •.. ,n k and division rings 01'···' Ok- Hopkins and Levitzki have proved the significant result that if R is a right Artinian ring then R is also right Noetherian, and J(R) is nilpotent. The following proposition is a consequence of this result.
-18- Proposition 1038. For a ring R, the following conditions are equivalent. (a) R is simple left Artinian ( b) R is simple right Art i n i a n ( c) R is simple and semisimple ( d) R =M (D), for some positive integer n and n some division ring D. RING OF FRACTIONS In the theory of commutative rings, Loc a l Ls a t i on at a multiplicative set plays a very important role. Most important is the idea of a quotient field, without which one can hardly imagine the study of integral domains. A very useful technique in commutative theory is the localisation at. a prime ideal, which ~educes many problems to the study of local rings and their maximal ideals. I-Iowever, this is not the case with non-commutative rings. Although the set of nonzero elements is a multiplicative set in any -domain, we have examples of domains which do not possess a division ring of quotientso It was in 1930, t~dt o. Ore characterised those non-commutative domains which possess division rings of fractions. In fact,
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: -16- Proposition 1 033. 'equiva len
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 and 68: -64- Theoren12.46. Let R be a pri.m
Page 69 and 70: Chapter-3 EXTENSIONS AND RINGS WITH
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-68- If S is a finite normalising e
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-70- regular element of R is regula
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-72- Lemma 3.110 Let P be a prime i
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-74- Now, if x ( P, then xS = Sx is
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-76- so that Sg = gS $ P. Since S i
Page 81 and 82:
-78- is an r' E R such that Equa t
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-80- As in [12, lemma 1 03]. Lemma
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-82- Examples 3023 (1) In [13J, a c
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-84- each j which is not possible.
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-86- Thus the ideal T(z+ux)T is not
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-88- in the generating set of I suc
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-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 [~~
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-9()- Remark 4 05. It can be seen t
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-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
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Lemma 4.28. Suppose P and Q are max
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-106- and P is primeo Cons equen t
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-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)
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-118- [17J B.J. Muller Localisation
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Studies in Rings generalised Unique Factorisation Rings

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