Studies in Rings generalised Unique Factorisation Rings

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-15- are distinguished by many nice propertieso For instance, every vector space is a direct sum of one-dimensional subspaces. We view simple modules as analogous to one dimensional spaces, :and the corresponding analogoues to higher dimenaional vector spaces are the semisimple modules; modules which are direct sums of simple submodules. Definition 1.32 0 The socle of an R-module A is the sum of all simple submodules of A and is denoted by sac A. A is sernisimple i"f A = s oc A. In any ring R, it is easy to observe that soc (RR) is an ideal of R. Similarly soc (RR) is an ideal of R, but these two socles need not coincide in generalo However, there are rings in which these two coincide. For instance R = Mn(D), where n is a positive integer and 0 is a division ring. In case n = 2, the right idea Is 1 1 = [g gJ 1 2 = [g gJ are the simple right ideals and M 2(D) = 1 1 fB 1 2 • Similarly M 2 (D) = J l $ 3 2, where 3 1 = m~ J 2 = [g g] are the simple left ideals. We state a propositiono
-16- Proposition 1 033. 'equiva len t.. For any ring R, the following conditions are (a) All right R-modules are semisimple ( b) All left R-modules are semisimple ( c) RR is semisimple (d) RR is semisirnple Definition 1 0 34 . A ring satisfying the conditions of Proposition 1033 is callpd a semisimple ring. Definition 1.35. A module A is Artinian provided A satisfies the descending chain condition (DCC) on submodules, i.e., there does not exist a properly descending infinite chain of 5ubmodules of A. A ring R is called right (left) Artinian if and only if the right R-module RR (left R-module RR) is Artinian. If both conditions hold, R is called an Artinian ring. Remark 1 036. As in the case of Noetherian structures it is easy to observe that-A is Artinian if and only if Ala and 8 are
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: -14- Definition 1 029. An ideal P i
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
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
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-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
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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
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-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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