Prime ideal of product ring
WebSince I is a primary ideal with associated prime P, we have that I ≠ R and P is a prime ideal. By the definition of primary ideal, for any a, b ∈ R such that a b ∈ I and a ∉ I, then b n ∈ P for some positive integer n. WebLet A (different from the zero ring) be a commutative ring with units. Suppose ring A has exactly one prime ideal. In this case, every element a of ring A is either an arithmetic unit or a nilpotent element. Prove it. Question: Let A (different from the zero ring) be a commutative ring with units. Suppose ring A has exactly one prime ideal.
Prime ideal of product ring
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Webfollows that an ideal p in the arbitrary ring R is a prime ideal in R if and only if aRb 0(p) implies that a--0(p) or b ( 0(). ... although stated for the product of two ideals, implies that … WebFeb 17, 2024 · The reason is that since P 1 and P 2 are proper, let a 1 ∈ R 1 ∖ P 1 and a 2 ∈ R 2 ∖ P 2. Then ( a 1, 0) ⋅ ( 0, a 2) = ( 0, 0) ∈ P 1 × P 2, but neither factor is in the ideal. …
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Webexistence of a prime ideal P disjoint from S such that Rx⊆P. As every nonzero prime ideal contains a prime element, P∩S= ∅implies that Pis the zero ideal, and so x= 0. Thus, every …
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