In what follows, R is the set of real numbers.
The following theorem is known about the ideals in a ring R.
A maximal ideal in a ring R is a prime ideal. Which of the following may be deduced as true from this theorem? Give a short explanation of each of your choices.

a. This theorem shows that if an ideal is prime, it must be maximal.
b. If an ideal is not maximal, then it must be prime.
c. If an ideal is not prime, then it cannot be maximal.

2. Originally Posted by ilovemath88

In what follows, R is the set of real numbers.
The following theorem is known about the ideals in a ring R.
A maximal ideal in a ring R is a prime ideal. Which of the following may be deduced as true from this theorem? Give a short explanation of each of your choices.

a. This theorem shows that if an ideal is prime, it must be maximal.
b. If an ideal is not maximal, then it must be prime.
c. If an ideal is not prime, then it cannot be maximal.

Try writing the theorem as an if then statement

If an Ideal in a ring is maximal, then it is a prime ideal

If P then Q.

a. is the inverse if Q then P this is not true.
b. not p implies q this is false
C. is the contrapositive not Q implies not P

This is true.

Good luck.