nLab anti-ideal

Idea

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

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

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

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

1. $0 \neq A$

2. $r_1 + r_2 \,\in\, A \;\;\;\;\;\Rightarrow\;\;\;\;\; r_1 \in A \;\;\text{or}\;\; r_2 \in A$

3. $r_1 \cdot r_2 \,\in\, A \;\;\;\;\;\Rightarrow\;\;\;\;\; r_1 \in A \;\;\text{and}\;\; r_2 \in A$.

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

References

• Anne Sjerp Troelstra, Dirk van Dalen, Def. 3.6 on p. 402 of: Constructivism in Mathematics – An introduction, Volume II, Studies in Logic and the Foundations of Mathematics 123: North Holland (1988) [ISBN:9780444703583]

• V. K. Kharchenko: Automorphisms and Derivations of Associative Rings, Mathematics and its Applications 69, Springer (1991) [doi:10.1007/978-94-011-3604-4]