Discussion
Consider the following statements :
I : A subring of a unique factorization domain is a unique factorization domain.
II : Let R be a unique factorization domain and \(d = \gcd(a, b)\) for \(a, b \in R\). Then \((d) = (a) + (b)\).
Which is correct ?
Answer: D
Neither I nor II is true.
I fails: the field C is trivially a UFD, but its subring \(\mathbf{Z}[\sqrt{-5}]\) is not, since \(6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})\) are two different factorisations into irreducibles.
II fails: in the UFD \(\mathbf{Z}[x]\), \(\gcd(2, x) = 1\), but the ideal \((2) + (x)\) consists of polynomials with even constant term and does not contain 1. The equality \((d) = (a) + (b)\) holds in a principal ideal domain, not in every UFD.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development