On rings which are sums of subrings and additive subgroups

Abstract

We prove that if a ring [Formula: see text] is a sum of its Jacobson radical subring [Formula: see text] and additive subgroup [Formula: see text] with [Formula: see text], then the ring [Formula: see text] is Jacobson radical, which answers a question asked by Kelarev and McConnell. Using this result we also show that for a semigroup [Formula: see text] the class of Jacobson radical rings is [Formula: see text]-sum closed if and only if [Formula: see text] is locally finite and all subgroups of [Formula: see text] are [Formula: see text]-groups.