Discussion
Consider the following statements :
(I) If R is a commutative ring with unity and \(f : R \to R\) be a ring homomorphism defined by \(f(a) = a^{2}\), then 1 + 1 = 0.
(II) If R is a commutative ring with unity and \(f : R \to R\) be a ring homomorphism defined by \(f(a) = a^{3}\), then 1 + 1 + 1 = 0.
Which of the following is true ?
Answer: A
Only (I) is true.
(I) If \(a \mapsto a^{2}\) is a ring homomorphism, it preserves addition, so \((1 + 1)^{2} = 1^{2} + 1^{2}\). The left side is \(4 \cdot 1\) and the right side is \(2 \cdot 1\), so \(2 \cdot 1 = 0\), that is \(1 + 1 = 0\). (In a commutative ring \((a + b)^{2} = a^{2} + 2ab + b^{2}\), and additivity forces \(2ab = 0\).)
(II) Applying the same idea to \(a \mapsto a^{3}\) gives \((1 + 1)^{3} = 1 + 1\), so \(8 \cdot 1 = 2 \cdot 1\), that is \(6 \cdot 1 = 0\). That does not force \(3 \cdot 1 = 0\). In \(\mathbf{Z}_{2}\), \(a^{3} = a\) for every element, so the cube map is the identity, a ring homomorphism, yet \(1 + 1 + 1 = 1 \neq 0\). So (II) is false, which rules out the options that include it.
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