Prime ideal

English

Etymology

By analogy with the notion of Template:M in number theory.

Noun

1. Template:Lb Any (two-sided) ideal $I$ such that for arbitrary ideals $P$ and $Q$, $PQ\subseteq I⟹P\subseteq I$ or $Q\subseteq I$.
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   • 1970 [Frederick Ungar Publishing], John R. Schulenberger (translator), Template:W, 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 $𝔬$.

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

Translations

• Italian: Template:T
• Polish: Template:T