Abstract
This paper deals with additive groups of rings in which all subrings are ideals. It is shown that if an abelian group supports only rings with this property, then all of them are commutative. This result is obtained for associative as well as not necessarily associative rings. Some previously known far from obvious results related to the mentioned groups are generalized and complemented. In particular, the classification of torsion-free abelian groups on which every not necessarily associative ring has the property that all its subrings are ideals is provided up to the structure of nil-groups.
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