This paper determines the structure of an associative ring $R$ when either all of its additive subgroups, all of its subrings, all of its (right) ideals, or all of its Lie ideals have finite additive index in $R$.
"Finite index conditions in rings." Rocky Mountain J. Math. 45 (4) 1177 - 1195, 2015. https://doi.org/10.1216/RMJ-2015-45-4-1177