6. Find two ideals I₁ and 12 of the ring Z such that a. I, U2 is not an ideal of Z. b. I₁ U2 is an ideal of Z.

14. Let be an ideal in a ring with unity . Prove that if then .

15. Let and be elements of a ring. Prove that the equation has a unique solution.

15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .

18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .

24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set is called the annihilator of in the ring .)

Let I1 and I2 be ideals of the ring R. Prove that the set I1I2=a1b1+a1b2+...+anbnaiI1,biI2,nZ+ is an ideal of R. The ideal I1I2 is called the product of ideals I1 and I2.

17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.

Let I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.

Exercises Let be an ideal of a ring , and let be a subring of . Prove that is an ideal of