# Introduction to Rings And Modules by C. Musili

By C. Musili

This booklet is a self-contained basic advent to jewelry and Modules, a subject constituting approximately 1/2 a center path on Algebra. The proofs are taken care of with complete info conserving the study room flavour. the total fabric together with workout is totally category verified. True/False statements are intended for a fast try out of realizing of the most textual content.

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ILLUS TRATION The Notation can 5 и 23 ϭ 5 и 8 ϭ 40 Ϫ5 и 23 ϭ Ϫ5 и 8 ϭ Ϫ40 Ϫ24 ϭ Ϫ͑24͒ ϭ Ϫ16 3͑Ϫ2͒3 ϭ 3͑Ϫ2͒͑Ϫ2͒͑Ϫ2͒ ϭ 3͑Ϫ8͒ ϭ Ϫ24 We next extend the definition of an to nonpositive exponents. 2 Exponents and Radicals 17 If m and n are positive integers, then m factors of a ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ aman ϭ a и a и a и и и и и a и a и a и a и и и и и a. n factors of a Since the total number of factors of a on the right is m ϩ n, this expression is equal to amϩn; that is, aman ϭ amϩn. We can extend this formula to m Յ 0 or n Յ 0 by using the definitions of the zero exponent and negative exponents.

11 (a) ͑Ϫ5͉͒ 3 Ϫ 6 ͉ (b) ͉ Ϫ6 ͉͑͞Ϫ2͒ (c) ͉ Ϫ7 ͉ ϩ ͉ 4 ͉ (e) t is not less than 5. 12 (a) ͑4͉͒ 6 Ϫ 7 ͉ (b) 5͉͞ Ϫ2 ͉ (c) ͉ Ϫ1 ͉ ϩ ͉ Ϫ9 ͉ (b) y is nonnegative. 7 Ϫ 23 ͉ (c) ͉ 51 Ϫ 13 ͉ Exer. 15 – 18: The given numbers are coordinates of points A, B, and C, respectively, on a coordinate line. Find the distance. (a) d(A, B) (b) d(B, C ) (c) d(C, B) (d) d(A, C ) 15 3, 7, Ϫ5 16 Ϫ6, Ϫ2, 4 17 Ϫ9, 1, 10 18 8, Ϫ4, Ϫ1 Exer. 41–42: Approximate the real-number expression to four decimal places. 64 (b) 3 Exer.

Polynomials Example Leading coefficient Degree 3x 4 ϩ 5x 3 ϩ ͑Ϫ7͒x ϩ 4 x 8 ϩ 9x 2 ϩ ͑Ϫ2͒x Ϫ5x2 ϩ 1 7x ϩ 2 8 3 1 Ϫ5 7 8 4 8 2 1 0 By definition, two polynomials are equal if and only if they have the same degree and the coefficients of like powers of x are equal. If all the coefficients of a polynomial are zero, it is called the zero polynomial and is denoted by 0. However, by convention, the degree of the zero polynomial is not zero but, instead, is undefined. If c is a nonzero real number, then c is a polynomial of degree 0.