nLab anti-ideal

Redirected from "antiideals".

Context

Algebra

Constructivism, Realizability, Computability

Contents

Idea

Where a (right) ideal in a magma (M,)(M, \cdot) is a subset IMI \subset M which “absorbs” elements, in that with iIi \in I and mMm \in M also the product imIi \cdot m \in I,

so a (right) anti-ideal is a subset AMA \subset M which “repels” elements, in that the only way that aAa \in A and mMm \in M have product amAa \cdot m \in A is if also mAm \in A (e.g. Kharchenko 1991 p. 190).

Analogously for left- and two-sided (anti-)ideals.

For the case of rings (R,,+)(R, \cdot, +) further conditions on the additive operation are imposed (e.g. Troelstra & van Dalen 1988, Def. 3.6 on p 402): a subset ARA \subset R is a two-sided anti-ideal of RR if:

  1. 0A0 \neq A

  2. r 1+r 2Ar 1Aorr 2Ar_1 + r_2 \,\in\, A \;\;\;\;\;\Rightarrow\;\;\;\;\; r_1 \in A \;\;\text{or}\;\; r_2 \in A

  3. r 1r 2Ar 1Aandr 2Ar_1 \cdot r_2 \,\in\, A \;\;\;\;\;\Rightarrow\;\;\;\;\; r_1 \in A \;\;\text{and}\;\; r_2 \in A.

See also at anti-subalgebra the example of anti-ideals.

References

Last revised on October 1, 2024 at 19:50:07. See the history of this page for a list of all contributions to it.