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This paper presents new results concerning the structure of 
$\text{SI}$-groups and refines and purifies the results obtained in this field by Shalom Feigelstock [‘Additive groups of rings whose subrings are ideals’, Bull. Aust. Math. Soc.55 (1997), 477–481]. The structure theorem describing torsion-free 
$\text{SI}$-groups is proved in the associative case. Numerous examples of 
$\text{SI}$-groups are given. Some inconsistencies in Feigelstock’s article are noted and corrected.
