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In this paper we extend the concepts of two sided ideal and right quasi-duo ring. These ideals and rings are called totally fully invariant and right strongly quasi-duo, respectively. Right strongly quasiduo rings are always right quasi-duo. We investigate the properties of these rings and ideals and show among other things that right strongly quasi-duo rings are classical, directly finite and co-hopfian while right quasi duo, right duo or commutative rings do not have necessarily these properties.

AMS Subject Classification: 13B22; 13B99.