Question:
Let \( {R}[X^2, X^3] \) be the subring of \( {R}[X] \) generated by \( X^2 \) and \( X^3 \). Consider the following statements:
1. The ring \( {R}[X^2, X^3] \) is a unique factorization domain.
2. The ring \( {R}[X^2, X^3] \) is a principal ideal domain.
Which one of the following is correct?
Let \( {R}[X^2, X^3] \) be the subring of \( {R}[X] \) generated by \( X^2 \) and \( X^3 \). Consider the following statements:
1. The ring \( {R}[X^2, X^3] \) is a unique factorization domain.
2. The ring \( {R}[X^2, X^3] \) is a principal ideal domain.
Which one of the following is correct?
2. The ring \( {R}[X^2, X^3] \) is a principal ideal domain.
Which one of the following is correct?
Show Hint
For domain-related problems, verify integral closure for UFD and the ideal structure for PID.
Updated On: Feb 1, 2025
- 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
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Analyzing the unique factorization domain (UFD).
The subring \( {R}[X^2, X^3] \) is not a unique factorization domain because it is not integrally closed. This property is essential for UFDs.
Step 2: Analyzing the principal ideal domain (PID).
The subring \( {R}[X^2, X^3] \) is not a principal ideal domain because it is not a free polynomial ring in one variable and hence does not satisfy the condition for every ideal to be principal.
Step 3: Conclusion.
Both statements are false. The correct answer is \( {(4)} \).
Was this answer helpful?
0
0
Top GATE MA Mathematics Questions
- Consider the following condition on a function \( f : {C} \to {C} \): \[ |f(z)| = 1 \quad {for all } z \in {C} { such that } \operatorname{Im}(z) = 0. \] Which one of the following is correct?
- Let \( C \) be the ellipse \( \{ z \in \mathbb{C} : |z - 2| + |z + 2| = 8 \} \) traversed counter-clockwise. The value of the contour integral \[ \int_C \frac{z^2 \, dz}{z^2 - 2z + 2} \] is equal to:
- Let \( X \) be a topological space and \( A \subseteq X \). Given a subset \( S \) of \( X \), let \( {int}(S), \partial S, \) and \( \overline{S} \) denote the interior, boundary, and closure, respectively, of the set \( S \). Which one of the following is NOT necessarily true?
Consider the following limit: $ \lim_{\epsilon \to 0} \frac{1}{\epsilon} \int_{0}^{\infty} e^{-x / \epsilon} \left( \cos(3x) + x^2 + \sqrt{x + 4} \right) dx. $
Which one of the following is correct?- Given a prime number \( p \), let \( n_p(G) \) denote the number of \( p \)-Sylow subgroups of a finite group \( G \). Which one of the following is TRUE for every group \( G \) of order \( 2024 \)?
View More Questions
Top GATE MA Product of Matrices Questions
- Consider the following condition on a function \( f : {C} \to {C} \): \[ |f(z)| = 1 \quad {for all } z \in {C} { such that } \operatorname{Im}(z) = 0. \] Which one of the following is correct?
- Given a prime number \( p \), let \( n_p(G) \) denote the number of \( p \)-Sylow subgroups of a finite group \( G \). Which one of the following is TRUE for every group \( G \) of order \( 2024 \)?
- Let \( A \in M_2({C}) \) be given by \[ A = \begin{bmatrix} 0 & 2
2 & 0 \end{bmatrix}. \] Let \( T: M_2({C}) \to M_2({C}) \) be the linear transformation given by \( T(B) = AB \). The characteristic polynomial of \( T \) is: - For the initial value problem \[ \frac{dy}{dx} = f(x, y), \quad y(x_0) = y_0, \] generate approximations \( y_n \) to \( y(x_n) \) using the recursion formula \[ y_n = y_{n-1} + a k_1 + b k_2, \] where \[ k_1 = h f(x_{n-1}, y_{n-1}), \quad k_2 = h f(x_{n-1} + \beta h, y_{n-1} + \beta k_1). \] Which one of the following choices of \( a, b, \alpha, \beta \) gives the Runge-Kutta method of order 2?
- Let \( u = u(x, t) \) be the solution of \[ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}, \quad 0<x<1, \, t>0, \] with boundary conditions \( u(0, t) = u(1, t) = 0 \) and initial condition \( u(x, 0) = \sin(\pi x) \). Define \[ g(t) = \int_0^1 u^2(x, t) \, dx. \] Which one of the following is correct?
View More Questions
Top GATE MA Questions
- No. of group homomorphism from Z4 to S4
- Number of finite non-isomorphic groups with exactly 3 conjugacy classes_____.
- f(x,y)=\(\left\{\begin{matrix} 1-cos(\frac{x^2}{y^2}) \\ \sqrt{x^2+y^2} \end{matrix}\right.\) ;if y≠0 & n∈R, then
- If ‘=’ denotes increasing order of intensity, then the meaning of the words [drizzle — rain — downpour] is analogous to [ — quarrel — feud]. Which one of the given options is appropriate to fill the blank?
- Statements: 1. All heroes are winners.
2. All winners are lucky people. Inferences: I. All lucky people are heroes.
II. Some lucky people are heroes.
III. Some winners are heroes. Which of the above inferences can be logically deduced from statements 1 and 2?
View More Questions