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Ideal criteria and intersections of ideals
Statement
Ideal criteria and intersections of ideals.
A nonempty subset is a two-sided ideal exactly when it is closed under and under for all , . Any intersection of two-sided ideals is a two-sided ideal, with the empty intersection equal to .
Facts & Assumptions
Given: A ring and a subset .
A two-sided ideal is an additive subgroup closed under left and right multiplication by ring elements (Left, right and two-sided ideals).
A nonempty subset closed under is a subgroup (One-step subgroup test: a nonempty is a subgroup iff for all ; the identity and the inverses of are then those of ).
A ring has additive inverses and distributive multiplication (Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides).
Proof
Closure under is exactly the additive subgroup criterion, and the two absorption conditions then give the ideal criterion.
An intersection has subtraction closure and both absorption properties because each member ideal has them; for the empty family the intersection is .
The criterion and intersection assertion follow.
Depends on
- Left, right and two-sided ideals
- One-step subgroup test: a nonempty $H \subseteq G$ is a subgroup iff $gh^{-1} \in H$ for all $g, h \in H$; the identity and the inverses of $H$ are then those of $G$
- Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides
Used by
- A ring with finitely many ideals of zero intersection whose quotients are Noetherian rings is Noetherian Corollary
- The zero ideal of ℤ is prime but not maximal Counterexample
- The classical vanishing ideal Definition
- The ideal generated by a subset and principal ideals Definition
- 2ℤ is an ideal of ℤ but is not a subring under the library's unital convention Example
- Vanishing sets and vanishing ideals form a contravariant Galois connection Example
- An ideal maximal among the non-finitely-generated ideals is prime Lemma
- If S is a subring and I is an ideal of R, then S+I is a subring, I is an ideal of S+I, and S∩ I is an ideal of S Lemma
- If some ideal is not finitely generated, there is one maximal among the ideals that are not Lemma
- The leading coefficients of the degree-n elements of an ideal of R[x], together with 0, form an ideal of R, and these ideals ascend with n Lemma
- In a commutative ring, (S) consists of finite sums ∑ rᵢ sᵢ, and (a)=Ra Theorem
- In a nonzero commutative ring, every proper ideal is contained in a maximal ideal Theorem
- R/M is a field if and only if M is a maximal ideal Theorem
- The sum and product of two-sided ideals are two-sided ideals Theorem
Dependency tree · two levels
13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Janssen and Lindsey, Rings with Inquiry, Ideals (standard reference, not scraped)