Phrase guide

prime ideal

noun · 2 senses · updated from the 2026-07-25 local source snapshot

Definitions and examples are grouped by meaning. Pronunciation, history, word forms, translations, descendants, synonyms, antonyms, derived terms, and related words appear whenever the source provides them.

Sound

Pronunciation

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1

noun

Meaning 1

Any (two-sided) ideal I such that for arbitrary ideals P and Q, PQ⊆I⟹P⊆I or Q⊆I.

Definition source: English Wiktionary via Wiktextract

Qualifier: ring theory

Topics: algebra, mathematics, sciences

Examples

  • 1970 [Frederick Ungar Publishing], John R. Schulenberger (translator), B. L. van der Waerden, Algebra, Volume 2, 2003, Springer, page 189, In the rings studied in Section 17.4 a nonzero prime ideal is divisible only by itself and by o on the basis of Axiom II; thus, in that section there are no lower prime ideals but o. Since every ideal a ne o is divisible by a prime ideal distinct from o (proof: from among all the divisors of a distinct from o choose a maximal one; since this ideal is maximal it is also prime), it follows that a cannot be quasi-equal to o.

Meaning relationships

Synonyms: none provided

Antonyms: none provided

2

noun

Meaning 2

In a commutative ring, a (two-sided) ideal I such that for arbitrary ring elements a and b, ab∈I⟹a∈I or b∈I.

Definition source: English Wiktionary via Wiktextract

No example sentence was provided for this meaning.

Meaning relationships

Synonyms: none provided

Antonyms: none provided

History

Etymology

By analogy with the notion of prime number in number theory.

Across languages

Translations

2 source translations are retained for this English entry.

  • Italian: ideale primo — (ring theory) type of ideal
  • Polish: ideał pierwszy — (ring theory) type of ideal