Difference between revisions of "Prime Ideals"
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| + | A ''prime ideal'' of a ring <math>R </math> is proper ideal <math>P </math> such that whenever <math>xy\in P </math> for <math>x,y \in R </math>, <math>x\in P </math> or <math>y \in P </math>. </br> | ||
| + | An ideal <math>P </math> is prime if and only if <math> R\backslash P</math> is closed under multiplication. | ||
| + | </br> | ||
| + | Maximal implies prime but not conversely. | ||
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| + | An ideal <math>P</math> is prime if and only if <math>R \P </math> is closed under multiplication. | ||
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| + | If <math>R</math> is a domain, <math>S</math> is a ring and <math>f : R \to S</math> is a ring homomorphism, then <math>\text{Ker}(f)</math> is a prime ideal. | ||
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| + | The ideal <math>I</math> is prime if and only if <math>R/I</math> is an [[Integral Domains|integral domain]]. | ||
Revision as of 20:46, 17 January 2023
- return to Algebra main page
A prime ideal of a ring \(R \) is proper ideal \(P \) such that whenever \(xy\in P \) for \(x,y \in R \), \(x\in P \) or \(y \in P \).
An ideal \(P \) is prime if and only if \( R\backslash P\) is closed under multiplication.
Maximal implies prime but not conversely.
An ideal \(P\) is prime if and only if \(R \P \) is closed under multiplication.
If \(R\) is a domain, \(S\) is a ring and \(f : R \to S\) is a ring homomorphism, then \(\text{Ker}(f)\) is a prime ideal.
The ideal \(I\) is prime if and only if \(R/I\) is an integral domain.
