prime ideal

Let $R$ be a ring. A two-sided proper ideal $\\mathfrak{p}$ of a ring $R$ is called a prime ideal if the following equivalent conditions are met:

1.
If $I$ and $J$ are left ideals and the product of ideals $I J$ satisfies $IJ\\subset\\mathfrak{p}$ , then $I\\subset\\mathfrak{p}$ or $J\\subset\\mathfrak{p}$ .
2.
If $I$ and $J$ are right ideals with $IJ\\subset\\mathfrak{p}$ , then $I\\subset\\mathfrak{p}$ or $J\\subset\\mathfrak{p}$ .
3.
If $I$ and $J$ are two-sided ideals with $IJ\\subset\\mathfrak{p}$ , then $I\\subset\\mathfrak{p}$ or $J\\subset\\mathfrak{p}$ .
4.
If $x$ and $y$ are elements of $R$ with $xRy\\subset\\mathfrak{p}$ , then $x\\in\\mathfrak{p}$ or $y\\in\\mathfrak{p}$ .

$R/\\mathfrak{p}$ is a prime ring if and only if $\\mathfrak{p}$ is a prime ideal. When $R$ is commutative with identity, a proper ideal $\\mathfrak{p}$ of $R$ is prime if and only if for any $a,b\\in R$ , if $a\\cdot b\\in\\mathfrak{p}$ then either $a\\in\\mathfrak{p}$ or $b\\in\\mathfrak{p}$ . One also has in this case that $\\mathfrak{p}\\subset R$ is prime if and only if the quotient ring $R/\\mathfrak{p}$ is an integral domain.

单词	PrimeIdeal
释义	prime ideal Let $R$ be a ring. A two-sided proper ideal $\\mathfrak{p}$ of a ring $R$ is called a prime ideal if the following equivalent conditions are met: 1. If $I$ and $J$ are left ideals and the product of ideals $I J$ satisfies $IJ\\subset\\mathfrak{p}$ , then $I\\subset\\mathfrak{p}$ or $J\\subset\\mathfrak{p}$ . 2. If $I$ and $J$ are right ideals with $IJ\\subset\\mathfrak{p}$ , then $I\\subset\\mathfrak{p}$ or $J\\subset\\mathfrak{p}$ . 3. If $I$ and $J$ are two-sided ideals with $IJ\\subset\\mathfrak{p}$ , then $I\\subset\\mathfrak{p}$ or $J\\subset\\mathfrak{p}$ . 4. If $x$ and $y$ are elements of $R$ with $xRy\\subset\\mathfrak{p}$ , then $x\\in\\mathfrak{p}$ or $y\\in\\mathfrak{p}$ . $R/\\mathfrak{p}$ is a prime ring if and only if $\\mathfrak{p}$ is a prime ideal. When $R$ is commutative with identity, a proper ideal $\\mathfrak{p}$ of $R$ is prime if and only if for any $a,b\\in R$ , if $a\\cdot b\\in\\mathfrak{p}$ then either $a\\in\\mathfrak{p}$ or $b\\in\\mathfrak{p}$ . One also has in this case that $\\mathfrak{p}\\subset R$ is prime if and only if the quotient ring $R/\\mathfrak{p}$ is an integral domain.
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