On almost radical ideals of noncommutative rings

Let J(R) denote the Jacobson radical of a commutative ring. In Khashan the notion of a J-ideal was introduced and studied. In khashanandece Khashan et al. introduced the concept of weakly J-ideals as a new generalization of J-ideals. In groenewald and groenerwaldweakly it was shown that many of the results are special cases of a more general situation. In TekirEce the notion of an almost n-ideal was introduced and studied. In this note we define almost ρ-ideals for a special radical ρ. If ρ is the prime radical then we have the almost n-radical for noncommutative rings. We prove amongst other results that an ideal I of a ring is an almost ρ-ideal if and only if I/I² is a weakly ρ-ideal of R. We also investigate rings for which every ideal is an almost radical ideal for a special radical ρ.

On almost radical ideals of noncommutative rings Keywords:

Special radical ρ , ρ-ideal , weakly-ρ-ideal , almost-ρ-ideal , fully almost-ρ-ring