On radical ideals of non-commutative rings

Let [Formula: see text] denote the Jacobson radical of a commutative ring [Formula: see text]. In [H. A. Khashan and A. B. Bani-Ata, [Formula: see text]-ideals of commutative rings, Int. Electron. J. Algebra 29 (2021) 148–164], the notion of J-ideals was introduced. If [Formula: see text] denotes the prime radical of a commutative ring then in [U. Tekir, S. Koc and K. H. Oral, [Formula: see text]-Ideals of commutative rings, Filomat 31(10) (2017) 2933–2941], the notion of an [Formula: see text] ideal of a commutative ring was introduced and studied. In this note, we show that these results are special cases of a more general situation. We define [Formula: see text]-ideals for a special radical [Formula: see text] and prove that most of the results of the above-mentioned papers are satisfied for non-commutative rings as a special case.