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Chapter 3, Rings

Chapter 3, RingsDe nitions and now have several examples of algebraic systems with addition and multiplication:Z;Zn;R;Mn(R);2Z=f2njn2Zg. We will write down a system of axioms whichincludes them nition, p. a nonempty setRwith two binary operations (usuallywritten as addition and multiplication) such that for alla; b; c2R,(1)Ris closed under addition:a+b2R.(2) Addition is associative: (a+b)+c=a+(b+c).(3) Addition is commutative:a+b=b+a.(4)Rcontains an additive identity element, calledzeroand usually denoted by 0 or0R:a+0=0+a=a.

The set of odd integers is not a ring. We can also work with matrices whose elements come from any ring we know about, such as Mn(Zr). Example. Let R = M2(Z2). This is a nite (16 elements) noncommutative ring with identity 1R = 10 01 and zero element 0R = 00 00 . We give an example to show it is noncommutative: 10 00 01 00 = 01 00

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