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Unformatted text preview: 24. Rings 24.1. Definitions and basic examples. Definition. A ring R is a set with two binary operations + (addition) and (multiplication) satisfying the following axioms: (A0) R is closed under addition (A1) addition is associative (A2) there exists 0 R s.t. a + 0 = 0 + a for all a R (A3) for every a R there exists- a R s.t. a + (- a ) = (- a ) + a = 0 (A4) addition is commutative (M0) R is closed under multiplication (M1) multiplication is associative (D1) distributivity on the left: ( a + b ) c = ac + bc for all a,b,c R (D2) distributivity on the right: c ( a + b ) = ca + cb for all a,b,c R Remark: The axioms (A0)-(A3) simply say that R is a group with respect to addition. Axiom (A4) says that the group ( R, +) is abelian. Definition. A ring R is called commutative if multiplication is commuta- tive, that is, ab = ba for all a,b R . Definition. A ring R is a called a ring with 1 (or a ring with unity ) if there exists 1 R s.t. a 1 = 1 a = a for all a R . Definition. A ring R is called a field if (i) R is commutative, (ii) R is a ring with 1 (iii) for every a R , with a 6 = 0, there exists a- 1 R such that a a- 1 = a- 1 a = 1 (iv) 1 6 = 0. Examples: 1. Fields. The examples of fields we know so far are Q , R , C and Z p where p is prime. 2. Commutative rings with 1, which are not fields. The examples we have seen so far are Z and Z n where n is non-prime. Another important class of examples is given by polynomial rings: Let R be a commutative ring with 1, and let R [ x ] denote the set of all polynomials with coefficients in R . One can think of R [ x ] as the set of formal 1 2 expressions a + a 1 x + ... + a n x n , with a i R . Addition and multiplication on R [ x ] are defined according to the usual algebra rules, e.g. ( a + a 1 x ) + ( b + b 1 x + b 2 x 2 ) = ( a + b ) + ( a 1 + b 1 ) x + b 2 x 2 and ( a + a 1 x ) ( b + b 1 x + b 2 x 2 ) = a b +( a b 1 + a 1 b ) x +( a b 2 + a 1 b 1 ) x 2 + a 1 b 2 x 3 . It is a rather long but straightforward calculation to check that R [ x ] with these operations becomes a ring with 1. The unity element of R [ x ] is the constant polynomial 1 (where 1 is the unity element of R )....
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This note was uploaded on 01/18/2012 for the course INFORMATIK 2011 taught by Professor Phanthuongcang during the Winter '11 term at Cornell University (Engineering School).
- Winter '11