By Donald S. Passman

First released in 1991, this e-book comprises the middle fabric for an undergraduate first direction in ring idea. utilizing the underlying subject matter of projective and injective modules, the writer touches upon quite a few points of commutative and noncommutative ring thought. particularly, a few significant effects are highlighted and proved. the 1st a part of the publication, known as "Projective Modules", starts with easy module idea after which proceeds to surveying a variety of particular sessions of jewelry (Wedderburn, Artinian and Noetherian jewelry, hereditary earrings, Dedekind domain names, etc.). This half concludes with an advent and dialogue of the thoughts of the projective size. half II, "Polynomial Rings", stories those jewelry in a mildly noncommutative atmosphere. a few of the effects proved comprise the Hilbert Syzygy Theorem (in the commutative case) and the Hilbert Nullstellensatz (for virtually commutative rings). half III, "Injective Modules", contains, particularly, a number of notions of the hoop of quotients, the Goldie Theorems, and the characterization of the injective modules over Noetherian jewelry. The publication comprises quite a few routines and an inventory of recommended extra analyzing. it truly is appropriate for graduate scholars and researchers drawn to ring conception.

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A. b. c. d. 6 12 15 18 153. If 11c – 7 = 8, what is the value of 33c – 21? a. b. c. 16 d. 24 e. 45 154. What value of x satisfies the equation ᎏxᎏ 2 a. + ᎏ16ᎏx = 4? ᎏ21ᎏ 4 1 ᎏ6ᎏ b. c. 3 d. 6 155. What value of b satisfies the equation b – ᎏ52ᎏ = –ᎏ23ᎏ? a. –ᎏ11ᎏ06 b. –3 c. d. 18 5 ᎏ1ᎏ 11 ᎏ8ᎏ 3 ᎏ53ᎏ 1 ᎏ16ᎏ –LINEAR EQUATIONS AND INEQUALITIES– 156. What value of c satisfies the equation 3c ᎏ4ᎏ – 9 = 3? a. 4 b. 12 c. 16 d. 20 157. What value of a satisfies the equation –ᎏ23ᎏa = –54 ? a. –81 b. 81 c.

Y y 10 10 8 8 6 6 4 4 2 2 –6 –9 3 –3 9 6 18 15 12 x –10 –2 –8 –6 –4 4 6 8 10 x –2 –4 a. b. c. d. 2 –2 –6 –4 –8 –6 –10 –8 yϾ8 yϽ8 xϽ8 xϾ8 –10 293. Which inequality is illustrated by the following graph? a. b. c. d. y–xϾ0 x–yϾ0 y–xՆ0 x–yՆ0 295. Which inequality is illustrated by the following y graph? 10 y 8 10 6 8 4 6 2 4 –10 –8 –6 –4 2 –2 4 6 8 10 x 2 –2 –4 –10 –8 –6 –4 2 –2 –2 –6 –4 –8 –6 –10 –8 a. b. c. d. 46 x+yՅ2 x–yՅ2 x – y Յ –2 x + y Յ –2 –10 a. b. c. d. ᎏ13ᎏx + 2y Ͼ –1 x +2y Ն –3 x + 6y Ͼ –1 ᎏ13ᎏx + 2 Ͻ –1 4 6 8 10 x –LINEAR EQUATIONS AND INEQUALITIES– 296.

C. d. e. (6,–4) (–6,4) (–6,–4) (–4,6) (6,–3) 229. The point (2,–5) lies in which quadrant? a. b. c. d. coordinates are given by (|–x – 2|, –|–x – 1|) must lie in which quadrant? a. Quadrant I b. Quadrant II c. Quadrant III d. Quadrant IV C (6,4) A 231. For all real numbers x Ͻ –2, points whose Quadrant I Quadrant II Quadrant III Quadrant IV 230. For all nonzero real numbers x and y, points whose coordinates are given by (x2,(–y)2) lie in which quadrant? a. Quadrant I b. Quadrant II c. Quadrant III d.

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