**UCSD, MATH 103B**

**Excerpt:** ... e are nonzero b and c such that ab = 0 and ca = 0. (The text only dened zero divisors for ** commutative ring **s.) (a) Prove that every nonzero nilpotent element of a ring is a zero divisor. Suppose R is a ** commutative ring ** and suppose a, b R satisfy an = 0 and bk = 0. It can be shown that (a b)n+k = 0. (You do NOT need to do this.) (b) Prove: If R is a ** commutative ring **, then the nilpotent elements of R are an ideal. (c) It was shown in class that the only ideals in the ring M2 (R) of 2 2 real matrices are the trivial ones {0} and M2 (R). Use this to show that commutative is necessary in (b). 01 00 Hint: Look at a = and b = . 00 10 END OF EXAM ...