Mathematics · Ring Theory
All rings considered below are assumed to be associative and commutative with 1 ≠ 0. Further, all ring homomorphisms map 1 to 1. Consider the following statements about such a ring R: P1: R is isomorphic to the product of two rings R1 and R2. P2: ∃ r1, r2 ∈ R such that r12 = r1 ≠ 0 ≠ r2 = r22, r1r2 = 0 and r1 + r2 = 1. P3: ∃ ideals I1, I2 ⊆ R with R ≠ I1 ≠ (0) ≠ I2 ≠ R such that R = I1 + I2 and I1 ∩ I2 = (0). P4: ∃ a, b ∈ R with a ≠ 0 ≠ b such that ab = 0. Then, which of the following implications is/are ALWAYS TRUE?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
Share:
- 1P4 ⇒ P1
- 2P3 ⇒ P4
- 3P2 ⇒ P3
- 4P1 ⇒ P2✓
Correct answer
P1 ⇒ P2
Download Single Question PDF
Save this question and verified answer as a clean study sheet with live clickable link.
👥
Share with Friends & Study Groups
Challenge your peer aspirants on WhatsApp or Telegram to solve this question!
Online Test Series
Start Free Mock Test →Preparing for Mathematics?
Practice full-length mock tests with real exam timers, instant score analysis, and all-India ranking.
Chapter Practice
Practice Ring Theory MCQs →Want more practice questions on Ring Theory?
Solve complete topic-wise tests with detailed solutions on GkGet.
Related Practice Questions
Recommended practice questions in Mathematics · Ring Theory