Question

7. The following are equivalent: (a) R is prime; (b) a,b ? R and aRb = 0 imply a = 0 or b = 0; (c) the right annihilator of every nonzero right ideal of R is 0; (d) the left annihilator of every nonzero left ideal of R is 0.

          7. The following are equivalent:
(a) R is prime;
(b) a,b ? R and aRb = 0 imply a = 0 or b = 0;
(c) the right annihilator of every nonzero right ideal of R is 0;
(d) the left annihilator of every nonzero left ideal of R is 0.

Added by Mercedes M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Mauya Mitchell

01/17/2024

Step 1

If a ≠ 0 and b ≠ 0, then aRb ≠ 0, which contradicts our assumption. Show more…

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The following are equivalent: (a) R is prime; (b) a,b ∈ R and aRb = 0 imply a = 0 or b = 0; (c) the right annihilator of every nonzero right ideal of R is 0; (d) the left annihilator of every nonzero left ideal of R is 0.