Geodesic
The Hereditary Property in the Lower Radical Construction
Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 474-476

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All rings considered are associative. We show that if a homomorphically closed class P1 of rings is hereditary in the sense that every ideal of a ring in P1 is also in P1 , then the lower Kurosh radical construction terminates at P 3. This is an improvement on the result of Anderson, Divinsky, and Sulinski (3) showing that the lower radical construction terminates at P 2 provided P1 is homomorphically closed, hereditary, and contains all zero rings. Examples are given to show that the third step is actually attained in some constructions. 
Armendariz, E. P.; Leavitt, W. G. The Hereditary Property in the Lower Radical Construction. Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 474-476. doi: 10.4153/CJM-1968-044-3
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