Studies in Rings generalised Unique Factorisation Rings

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-7- Proposition loll. Let R be a right Noetherian ring and let S be a subring of M (R). n If S contains the 5ubring RI = {diagOnal (r,r ... ,r) IrE: R} of all scalar matrices, then S is right Noetherian. In particular M (R) is a right Noetherian ring. n Proof It is obvious that R is isomorphic to RI and M n (R) is generated as a right RI module by the standard n x n matrix unitso Since R' is right Noetherian and the number of e .. 1 5 is finite, Mn(R) is a Noetherian RI-module, 1.J I by corollary 1 09. As all right ideals of S are also right R'-submodules of M (R), we conclude that S is right Noetherian. n PRIME IDEALS It is well known that the prime ideals are the 'building blo~ks' of ideal theory in commutative rings. We recall that a proper ideal P in a commutative ring is said to be prime if whenever we have two elements a and b in R such tha t ab E: P, i t follows tha t either a e P or b € P; equivalently P is prime if and only if RIp is a domain.
-8- In non-commutative rings, it turns out that it is not a good idea to concentrate on prime ideals P such that RIp is a domain (ab E P implies a f: P or b E p). In f a c t , there are many non-commutative rings with no factor rings which are domains. Thus the desirable thing is to give a more relaxed definition for prime idealso The key is to change the commutative definition by replacing products of elements by products of ideals which was first proposed by Krull in 1928. Definition 1.12. A prime ideal in a ring R is a proper ideal P of R such that whenever I and J are ideals of R with IJ ~ P, either I ~ P or J ~ P, P is said to be a completely prime ideal, if whenever a,b t R such that ab E P, either a c: P or b £ P. A prime ring is a ring in which 0 is a prime ideal and a domain is a ring in which 0 is a completely prime ideal. From part (c) of the following proposition it follows t hat in th e co mmuta t i v e case th e pri me i de a 15 and th e completely prime ideals coincide with the usual prime ideals and is a in non-commutative setting, every completely prime ideal prime ideal.
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: -6- If both conditions hold, R is s
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 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
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-60- RI~JGS yVITli [::NOUGH INVER~r
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-62- Th e o r ern 2.43. Let R be a
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-64- Theoren12.46. Let R be a pri.m
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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
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-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
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-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~) =
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-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
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REFERENCES [1] AoW. Chatters (1984)
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-118- [17J B.J. Muller Localisation
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