Multiplicative subgroups of finite index in a ring

Multiplicative subgroups of finite index in a ring
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by Vitaly Bergelson and Daniel B. Shapiro PDF

Proc. Amer. Math. Soc. 116 (1992), 885-896 Request permission

If $G$ is a subgroup of finite index in the multiplicative group of an infinite field $K$ then $G - G = K$. Similar results hold for various rings.

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