Fully invariant ideal

Definition
Note: This definition is structurally the same both for commutative and non-commutative rings.

Symbol-free definition
An ideal in a ring is termed fully invariant or a T-ideal if it is invariant under every endomorphism of the ring.

Definition with symbols
An ideal $$I$$ in a ring $$R$$ is termed fully invariant or a T-ideal in $$R$$, if, for any endomorphism $$\sigma$$ of $$R$$, the image of $$I$$ under $$\sigma$$ lies inside $$I$$.

Stronger properties

 * Verbal ideal

Weaker properties

 * Characteristic ideal
 * Normal ideal