Prime Ideals - Revision history

Kfagerstrom at 01:12, 8 March 2023

2023-03-08T01:12:03Z

2023-01-17T20:47:27Z

2023-01-17T20:46:48Z

2023-01-17T20:41:04Z

Created page with ":''return to Algebra main page"

New page

:''return to [[817 - Algebra#Prime Ideals| Algebra main page]]

← Older revision		Revision as of 01:12, 8 March 2023
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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]].		The ideal <math>I</math> is prime if and only if <math>R/I</math> is an [[Integral Domains\|integral domain]].
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		+	[[Category: Ring Theory]]

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 Maximal implies prime but not conversely.
-An ideal <math>P</math> is prime if and only if <math>R \P </math> is closed under multiplication.
+An ideal <math>P</math> is prime if and only if <math>R \backslash P </math> is closed under multiplication.
 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.
 The ideal <math>I</math> is prime if and only if <math>R/I</math> is an [[Integral Domains|integral domain]].

← Older revision		Revision as of 20:46, 17 January 2023
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	:''return to [[817 - Algebra#Prime Ideals\| Algebra main page]]		:''return to [[817 - Algebra#Prime Ideals\| Algebra main page]]
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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]].