Prove that if A and B are ideals such that A|B and B|A, then A = B.
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- Exercises If and are two ideals of the ring , prove that the set is an ideal of that contains each of and . The ideal is called the sum of ideals of and .15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .Find the principal ideal (z) of Z such that each of the following sums as defined in Exercise 8 is equal to (z). (2)+(3) b. (4)+(6) c. (5)+(10) d. (a)+(b) If I1 and I2 are two ideals of the ring R, prove that the set I1+I2=x+yxI1,yI2 is an ideal of R that contains each of I1 and I2. The ideal I1+I2 is called the sum of ideals of I1 and I2.
- Exercises Let be an ideal of a ring , and let be a subring of . Prove that is an ideal ofLabel each of the following statements as either true or false. The only ideal of a ring R that property contains a maximal ideal is the ideal R.27. If is a commutative ring with unity, prove that any maximal ideal of is also a prime ideal.