MATH145 Lecture Notes - Lecture 19: Commutative Ring

• School: University of Waterloo
• Department: Mathematics
• Course: MATH145
• Professor: Stephen New

Document Summary

A ring (with identity) is a set r with two distinct elements 0, 1 r and two binary operations. A, b, c r (a + b) + c = a + (b + c) A, b r a + b = b + a. A r a + 0 = a. A r b r a + b = 0. A, b, c r (ab)c = a(bc) A r a 1 = a and 1 a = a. A, b, c r a(b + c) = ab + ac and (a + b)c = ac + bc. A ring r is commutative when: is commutative. A eld is commutative ring r such that: every nonzero a r has an inverse under . 0 6= a r b r ab = 1. N is not a ring (axiom 4 does not hold) Z is not a eld (axiom 9 does not work)