Ideal

Prime ideal

A (two-sided) proper ideal is called a prime ideal iff for any , implies or ¹, ring i.e.

Historical note

Considering the original notion of an Ideal number, an ideal is the set of multiples of an ideal number . Therefore the above is equivalent to

i.e. is prime.

Note an ideal is prime iff where is prime or zero. The set of all prime ideals of a commutative ring is called its spectrum.

Properties

• Condition for a quotient commutative ring to be an integral domain (often this is used to prove primality)
• A maximal ideal in a commutative ring is prime
• ^D2

See also

• Krull dimension
• Relatively prime ideals
• Prime order of an ideal

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tidy | en | sembr

Footnotes

1. 2017. Contemporary abstract algebra, §14, p. 253 ↩