Abstract Algebra : Principal Ideals

Study concepts, example questions & explanations for Abstract Algebra

varsity tutors app store varsity tutors android store

Example Questions

Example Question #1 : Principal Ideals

Which of the following is an ideal of a ring?

Possible Answers:

Minimum Ideal

All are ideals of rings.

Multiplicative Ideal

Prime Ideal

Associative Ideal

Correct answer:

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 #2 : Principal Ideals

Which of the following is an ideal of a ring?

Possible Answers:

Associative Ideal

Maximal Ideal

None are ideals

Communicative Ideal

Minimal Ideal

Correct answer:

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 #3 : Principal Ideals

Which of the following is an ideal of a ring?

Possible Answers:

Proper Ideal

All are ideals

Associative Ideal

Minimal Ideal

Communicative Ideal

Correct answer:

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.

Learning Tools by Varsity Tutors