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De nition and Examples of Rings

LECTURE 14 Definition and Examples of a nonempty setRequipped with two operations and (more typicallydenoted as addition and multiplication) that satisfy the following conditions. For alla, b, c R:(1)Ifa Randb R, thena b R.(2)a (b c) = (a b) c(3)a b=b a(4)There is an element0 RinRsuch thata 0R=a , a R .(5)For eacha R, the equationa x= 0 Rhas a solution inR.(6)Ifa R, andb R, thenab R.(7)a (b c) = (a b) c.(8)a (b c) = (a b) (b c) ringis a ringRsuch that( )a b=b a , a, b R . with identityis a ringRthat contains an element1 Rsuch that( )a 1R= 1R a=a , a R .Let us continue with our discussion of Examples of ,Q,R, andCare all commutative Rings with an interval on the real line and letRdenote the set of continuous functionsf:I be given the structure of a commutative ring with identity by setting[f g](x)=f(x) +g(x)[f g](x)=f(x)g(x)0R function with constant value 01R function with constant value 1and then verifying that properties (1)-(10) the set of continuous functionsf:R Rsuch that 0f(x)dx <.

However, the ring Q of rational numbers does have this property. Definition 14.7. A division ring is a ring R with identity 1 R 6= 0 R such that for each a 6= 0 R in R the equations a x = 1 R and x a = 1 R have solutions in R. Note that we do not require a division ring to be commutative. Definition 14.8. A eld is a division ring with ...

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  Ring, Rational, Of rational

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