on almost radical ideals of noncommutative rings

Let j (r) denote the jacobson radical of a commu- tative ring r. in [8] the notion of a j -ideal was introduced and studied. in [7] khashan et al. introduced the concept of weakly j -ideals as a new generalization of j -ideals. in [4] and [5] it was shown that many of the results are special cases of a more general situation. in [1] 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-ideals 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.