Kwangil Koh
Abstract
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A ring R is called a prime ring [1] if and only if a·R·b = 0 implies that a = 0 or b = 0 for all a, b ϵ R. Hence if R is a prime ring and a is a non-zero element of R, a·R ≠ 0 and R·a ≠ 0. In the present note we prove that a prime ring with a maximal annihilator right ideal has no non-zero nil right or left ideal.
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A ring R is called a prime ring [1] if and only if a·R·b = 0 implies that a = 0 or b = 0 for all a, b ϵ R. Hence if R is a prime ring and a is a non-zero element of R, a·R ≠ 0 and R·a ≠ 0. In the present note we prove that a prime ring with a maximal annihilator right ideal has no non-zero nil right or left ideal.
Key concepts: Mathematics, Annihilator, Associated prime, Ideal (ethics), Reduced ring, Minimal ideal, Radical of a ring, Principal ideal ring