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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.(5) Every element ofRhas an additive inverse: for eacha, there exists anx2 Rsuchthata+x=0=x+a. We writex= a.(6)Ris closed under multiplication:ab2R.(7) Multiplication is associative: (ab)c=a(bc).

The zero ring is a subring of every ring. As with subspaces of vector spaces, it is not hard to check that a subset is a subring as most axioms are inherited from the ring. Theorem 3.2. Let S be a subset of a ring R. S is a subring of R i the following conditions all hold: (1) S is closed under addition and multiplication. (2) 0R 2 S.

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