By Carl Faith

VI of Oregon lectures in 1962, Bass gave simplified proofs of a few "Morita Theorems", incorporating principles of Chase and Schanuel. one of many Morita theorems characterizes whilst there's an equivalence of different types mod-A R::! mod-B for 2 jewelry A and B. Morita's resolution organizes principles so successfully that the classical Wedderburn-Artin theorem is a straightforward outcome, and furthermore, a similarity type [AJ within the Brauer team Br(k) of Azumaya algebras over a commutative ring okay involves all algebras B such that the corresponding different types mod-A and mod-B inclusive of k-linear morphisms are identical by way of a k-linear functor. (For fields, Br(k) involves similarity sessions of easy important algebras, and for arbitrary commutative okay, this is often subsumed below the Azumaya [51]1 and Auslander-Goldman [60J Brauer staff. ) various different circumstances of a marriage of ring concept and classification (albeit a shot gun wedding!) are inside the textual content. moreover, in. my try and extra simplify proofs, particularly to dispose of the necessity for tensor items in Bass's exposition, I exposed a vein of rules and new theorems mendacity wholely inside ring idea. This constitutes a lot of bankruptcy four -the Morita theorem is Theorem four. 29-and the foundation for it's a corre spondence theorem for projective modules (Theorem four. 7) urged through the Morita context. As a spinoff, this offers beginning for a slightly whole idea of easy Noetherian rings-but extra approximately this within the creation.

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**Example text**

13x - 2y213x + 2y2 36. 14x 2 - 5y22 40. 12x - 7212x + 72 42. 13x + 5y213x - 5y2 43. 12x + 3y + 4212x + 3y - 42 44. 15x + 2y + 3215x + 2y - 32 45. 1x + 121x - 121x 2 + 12 46. 1 y - 221 y + 221 y 2 + 42 22 Basic Concepts of Algebra CHAPTER R 50. 1x 3m - t 5n22 Synthesis Multiply. Assume that all exponents are natural numbers. 47. 1an + bn21an - bn2 51. 1x - 121x 2 + x + 121x 3 + 12 52. 312x - 122 - 142 53. 1x a - b2a + b 54. 1t m + n2m + n # 1t m - n2m - n 48. 1t a + 421t a - 72 55. 1a + b + c22 49.

Then divide by multiplying by the reciprocal of the denominator. EXAMPLE 6 1 1 + a b Simplify: . 1 1 + 3 a3 b 42 CHAPTER R Basic Concepts of Algebra Solution Method 1. The LCD of the four rational expressions in the numerator and the denominator is a3b3. 1 1 1 1 + + a a b b # a3b3 a3b3 Multiplying by 1 using = 1 1 1 1 a3b3 a3b3 3 + 3 3 + 3 a b a b 1 1 a + b 1a3b32 a b = 1 1 a 3 + 3 b1a3b32 a b = a2b3 + a3b2 b3 + a3 a2b21b + a2 = 1b + a21b2 - ba + a22 = a2b2 b2 - ba + a2 Factoring and removing b؉a a factor of 1: ؍1 b؉a Method 2.

5x|x … - 26 16. 5x|x 7 - 56 19. 5x|7 6 x6 ( 0 1 2 3 4 5 6 3 4 5 6 Ϫ10 Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 22. [ Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 ] 2 23. [ 24. 36. - 1 ʦ ޗ 38. 1 ʦ ޚ ޗ މ 40. ގ8 ޗ 41. ޗ8 ޚ 42. ޚ8 ގ 43. ޑ8 ޒ 44. ޚ8 ޑ 45. ޒ8 ޚ 46. ޑ8 މ Name the property illustrated by the sentence. 47. 3 + y = y + 3 49. - 3 # 1 = - 3 50. 41y - z2 = 4y - 4z ] Ϫ10 Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 ޒ 34. - 26 ʦ ޑ 48. 61xz2 = 16x2z ) ( 32. 1 ʦ ޒ 11 ʦޑ 5 39. 089 ) 1 33. - 37. 24 Write interval notation for the graph.