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On additive groups of associative and commutative rings

Abstract The characterization of (A)CR-groups studied by Shalom Feigelstock [‘Additive groups of commutative rings’, Quaest. Math. 23 (2000), 241–245] is complemented. In particular, the theorem characterizing mixed ACR-groups is proved and numerous new examples of (A)CR-group are given. Abelian groups A such that any ring with the additive group A is associative and commutative are partially described. Some new results for torsion-free and mixed TI-groups are obtained.