Using the Chinese Remainder Theorem

  1. Suppose I and J are ideals in a commutative ring R such that R=I+J. Prove that the map f:RR/I×R/J given by f(x)=(x+I,x+J) induces the isomorphism

    R/IJR/I×R/J.
  2. Prove that (Z/3Z)[x]/(x3x21)(Z/3Z)[x]/(x3+x+1). (Hint: Use part (1).)