GATE 2025 MA – Question 59
Rings are commutative associative with 1≠0 and unital homomorphisms. P1:R is a product of two nonzero rings. P2:R has nonzero idempotents r₁,r₂ with r₁r₂=0 and r₁+r₂=1. P3:nonzero proper ideals I₁,I₂ sum to R and intersect in (0). P4:R has nonzero zero divisors a,b. Which implications hold?
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Correct answer: (A) P1⇒P2; (B) P2⇒P3; (C) P3⇒P4
Explanation
In a product, (1,0),(0,1) give P2. Orthogonal complementary idempotents generate ideals Rr₁,Rr₂ that sum to R and intersect only in zero, proving P2⇒P3. For P3, products of nonzero a∈I₁,b∈I₂ lie in their zero intersection, proving P4. The converse fails in ℤ₄: 2×2=0 but only idempotents are 0,1, so no nontrivial product decomposition. **A,B,C**.