Abstract
We study rings [Formula: see text] which are sums of a subring [Formula: see text] and an additive subgroup [Formula: see text]. We prove that if [Formula: see text] is a prime radical ring and [Formula: see text] satisfies a polynomial identity, then [Formula: see text] is nilpotent modulo the prime radical of [Formula: see text]. Additionally, we show that if [Formula: see text] is a [Formula: see text] ring, then the prime radical of [Formula: see text] is nilpotent modulo the prime radical of [Formula: see text]. We also obtain a new condition equivalent to Koethe’s conjecture.
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