algebra abstract - robert b ash

... (−a )b = a( b) =−(ab)[0= 0b =(a+(−a) )b = ab+(−a )b, so (−a )b = −(ab); similarly, 0=a0=a (b +( b) ) = ab + a( b) , so a( b) =−(ab)] (3) (−1)(−1) = 1 [take a =1 ,b = −1 in (2)] (4) (−a)( b) =ab [replace b by ... Frontmatter Abstract < /b> Algebra:< /b> The Basic Graduate Year Robert < /b> B. Ash PREFACE This is a text for the basic graduate sequence in abstract < /b> algebra...

Ngày tải lên: 31/03/2014, 14:59

... owned the existing The Project Gutenberg EBook of History of the Comstock Patent Medicine Business and Dr. Morse's Indian Root Pills, by Robert < /b> B. Shaw This eBook is for the use of anyone anywhere ... Indian Root Pills Author: Robert < /b> B. Shaw Release Date: September 8, 2004 [EBook #13397] [Last updated: September 21, 2011] trade names and formulas. Dispute was inevitable u...

Ngày tải lên: 15/03/2014, 17:20

... result is obvious. Suppose neither a nor b is 0. Now 0 ≤ (ua ± vb) ·(ua ±vb) = (u ·u)a 2 ± 2ab(u · v)+ (v ·v )b 2 = b 2 a 2 ±2ab(u·v)+a 2 b 2 . Dividing by 2ab yields 0 ≤ ab±(u·v) or | u·v |≤ ab. Chapter ... divides b (a |b) if ∃! c ∈ R with ac = b. Theorem Suppose R is a domain and a, b ∈ (R − 0 ¯ ). Then the following are equivalent. 1) a ∼ b. 2) a |b and b| a. 3) aR = bR....

Ngày tải lên: 17/01/2014, 04:20

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Ngày tải lên: 18/02/2014, 21:20

... motions of an n-gon, but can also be described more abstractly in terms of two generators a (of order n) and b (of order 2) which satisfy the relation ba = a −1 b. We can write D n = {a i b j | 0 ≤ ... at: http://www.math.niu.edu/ ∼ beachy /abstract < /b> algebra/< /b> 24 CHAPTER 3. GROUPS 3.8 Cosets, Normal Subgroups, and Factor Groups The notion of a factor group is one of the most...

Ngày tải lên: 27/03/2014, 11:47

... G, b ∈ G with a · b = b ·a = e If a ∈ G, b ∈ G with a + b = b + a =0 ¯ (b is written as b = a −1 ). (b is written as b = −a). 4) If a, b ∈ G, then a · b = b · a. If a, b ∈ G, then a + b = b + ... and b. 2) G is a subgroup. In fact, it is the smallest subgroup containing a and b. It is called the subgroup generated by a and b. 3) Denote by (a, b) the smalles...

Ngày tải lên: 27/03/2014, 11:48

... Reduced Row-Echelon Form Contributed by Robert < /b> Beezer Solution [51] C13 x 1 + 2x 2 + 8x 3 − 7x 4 = −2 3x 1 + 2x 2 + 12x 3 − 5x 4 = 6 −x 1 + x 2 + x 3 − 5x 4 = −10 Contributed by Robert < /b> Beezer Solution ... x 2 + 5x 3 − 3x 4 = 1 Contributed by Robert < /b> Beezer Solution [52] C15 2x 1 + 3x 2 − x 3 − 9x 4 = −16 x 1 + 2x 2 + x 3 = 0 −x 1 + 2x 2 + 3x 3 + 4x 4 = 8 Contributed by Ro...

Ngày tải lên: 31/03/2014, 14:57

... r 1 + b 1 and s = s 1 + b 2 for some b 1 ,b 2 ∈ B. But rs =(r 1 + b 1 )(s 1 + b 2 )=r 1 s 1 + r 1 b 2 + b 1 s 1 + b 1 b 2 ∈ r 1 s 1 + B. Thus, rs ∈ r 1 s 1 + B, and hence, rs +B = r 1 s 1 + B. Therefore, ... =0or δ(r) <δ (b) . But since r = a − bq and B is an ideal, then r ∈ B. By the choice of b, it follows that r = 0. Therefore, a = bq, and a ∈ (b) . Hence,...

Ngày tải lên: 31/03/2014, 15:02