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lecture24

# lecture24 - 24 Rings 24.1 Definitions and basic examples...

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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

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2 expressions a 0 + 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 0 + a 1 x ) + ( b 0 + b 1 x + b 2 x 2 ) = ( a 0 + b 0 ) + ( a 1 + b 1 ) x + b 2 x 2 and ( a 0 + a 1 x ) · ( b 0 + b 1 x + b 2 x 2 ) = a 0 b 0 +( a 0 b 1 + a 1 b 0 ) x +( a 0 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 ). 3. Commutative rings without 1. The basic examples are the rings n Z where n 2 is a fixed integer. 4. Noncommutative rings. The basic examples are the matrix rings Mat n ( F ) where F is some field and n 2.
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