Let

be an endomorphism and I a

-ideal of a ring R. Pearson and Stephenson called I a

-semiprime ideal if whenever A is an ideal of R and m is an integer such that

for all

, then

, where

is an automorphism, and Hong et al. called I a

-rigid ideal if

implies a

for

. Notice that R is called a

-semiprime ring (resp., a

-rigid ring) if the zero ideal of R is a

-semiprime ideal (resp., a

-rigid ideal). Every

-rigid ideal is a

-semiprime ideal for an automorphism

, but the converse does not hold, in general. We, in this paper, introduce the quasi

-rigidness of ideals and rings for an automorphism

which is in between the

-rigidness and the

-semiprimeness, and study their related properties. A number of connections between the quasi

-rigidness of a ring R and one of the Ore extension

of R are also investigated. In particular, R is a (principally) quasi-Baer ring if and only if

is a (principally) quasi-Baer ring, when R is a quasi

-rigid ring.