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 $I$.

Relation with other properties

Stronger properties

• Verbal ideal

Weaker properties

• Characteristic ideal
• Normal ideal