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In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal. Primitive ideals are prime, and prime ideals are both primary and semiprime. ==Prime ideals for commutative rings== An ideal of a commutative ring is prime if it has the following two properties: * If and are two elements of such that their product is an element of , then is in or is in , * is not equal to the whole ring . This generalizes the following property of prime numbers: if is a prime number and if divides a product of two integers, then divides or divides . We can therefore say :A positive integer is a prime number if and only if the ideal is a prime ideal in . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「prime ideal」の詳細全文を読む スポンサード リンク
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