GATE MA concept:principal-ideal-domains PYQs & Answers

Questions 1–3 of 3. Imported answers require independent review unless marked reviewed.

Let

be the subring of

generated by

and

. Consider the following statements.

I. The ring

is a unique factorization domain.

II. The ring

is a principal ideal domain.

Which one of the following is correct?

Your answerChoose one option.

Both I and II are TRUE

I is TRUE and II is FALSE

I is FALSE and II is TRUE

Both I and II are FALSE

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Let

, where

denotes the set of rational numbers and

denotes the set of integers. For

, let

denote the ideal generated by

Which of the following statements is/are correct?

Your answerSelect all correct options. You can choose more than one.

is an irreducible element in

, then

is a prime ideal in

is a unique factorization domain

is a prime ideal in

is NOT a principal ideal domain

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Consider the following statements:

I. The ring

is a unique factorization domain.

II. The ring

is a principal ideal domain.

III. In the polynomial ring

, the ideal generated by

is a maximal ideal.

IV. In the polynomial ring

, the ideal generated by

is a prime ideal.

(

denotes the set of all integers,

denotes the set of all integers modulo

, for any positive integer

)

Which of the above statements are TRUE?

Your answerChoose one option.

I, II and III only

I and II only

I, II and IV only

II and III only

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GATE MA concept:principal-ideal-domains PYQs

How to practice concept:principal-ideal-domains

Question 15

Question 45

Question 35