College Algebra Exam Review 254

College Algebra Exam Review 254 - Chapter 6 Rings 6.1. A...

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Unformatted text preview: Chapter 6 Rings 6.1. A Recollection of Rings We encountered the definitions of rings and fields in Section 1.11. Let us recall them here for convenience. Definition 6.1.1. A ring is a nonempty set R with two operations: addition, denoted here by C, and multiplication, denoted by juxtaposition, satisfying the following requirements: (a) Under addition, R is an abelian group. (b) Multiplication is associative. (c) Multiplication distributes over addition: a.b C c/ D ab C ac , and .b C c/a D ba C ca for all a; b; c 2 R. A ring is called commutative if multiplication is commutative, ab D ba for all elements a; b in the ring. Recall that a multiplicative identity in a ring is an element 1 such that 1a D a1 D a for all elements a in the ring. An element a in a ring with multiplicative identity 1 is a unit or invertible if there exists an element b such that ab D ba D 1. Some authors include the the existence of a multiplicative identity in the definition of a ring, but as this requirement excludes many natural examples, we will not follow this practice. Let’s make a few elementary deductions from the ring axioms: Note that the distributive law a.b Cc/ D ab Cac says that the map La W b 7! ab is a group homomorphism of .R; C/ to itself. It follows that La .0/ D 0 and La . b / D La .b/ for any b 2 R. This translates to a 0 D 0 and a. b / D ab . Similarly, Ra W b 7! ba is a group homomorphism of .R; C/ to itself, and, consequently, 0 a D 0, and . b /a D b a. For n 2 Z and a 2 R, since nb is the n–th power of b in the abelian group .R; C/, we also have La .nb/ D nLa .b/; that is, a.nb/ D n.ab/. Similarly, Ra .nb/ D nLa .b/; that is, .nb/a D n.ba/. 264 ...
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