# Abstract Algebra : Principal Ideals

## Example Questions

### Example Question #3 : Abstract Algebra

Which of the following is an ideal of a ring?

Minimum Ideal

Multiplicative Ideal

Prime Ideal

All are ideals of rings.

Associative Ideal

Prime Ideal

Explanation:

When dealing with rings there are three main ideals

Proper Ideal: When  is a commutative ring, and  is a non empty subset of  then,  is said to have a proper ideal if both the following are true.

and

Prime Ideal: When  is a commutative ring,  is a prime ideal if

is true and

Maximal Ideal: When  is a commutative ring, and  is a non empty subset of  then,  has a maximal ideal if all ideal  are

Looking at the possible answer selections, Prime Ideal is the correct answer choice.

### Example Question #1 : Principal Ideals

Which of the following is an ideal of a ring?

Associative Ideal

Maximal Ideal

None are ideals

Minimal Ideal

Communicative Ideal

Maximal Ideal

Explanation:

When dealing with rings there are three main ideals

Proper Ideal: When  is a commutative ring, and  is a non empty subset of  then,  is said to have a proper ideal if both the following are true.

and

Prime Ideal: When  is a commutative ring,  is a prime ideal if

is true and

Maximal Ideal: When  is a commutative ring, and  is a non empty subset of  then,  has a maximal ideal if all ideal  are

Looking at the possible answer selections, Maximal Ideal is the correct answer choice.

### Example Question #5 : Abstract Algebra

Which of the following is an ideal of a ring?

Proper Ideal

Communicative Ideal

All are ideals

Associative Ideal

Minimal Ideal

Proper Ideal

Explanation:

When dealing with rings there are three main ideals

Proper Ideal: When  is a commutative ring, and  is a non empty subset of  then,  is said to have a proper ideal if both the following are true.

and

Prime Ideal: When  is a commutative ring,  is a prime ideal if

is true and

Maximal Ideal: When  is a commutative ring, and  is a non empty subset of  then,  has a maximal ideal if all ideal  are

Looking at the possible answer selections, Prime Ideal is the correct answer choice.