Generating Ring from its Group of Units

Question

Say $E$ is the group of units of a ring $R$. Is it possible to generate $R$ from $E$? Analogous to how $R$ is generated by its fundamental units.

My hunch says all elements of $R$ should be linear combinations of $E$; am I completely off base?

Answer 1

No, consider any polynomial ring over a field. The group of units will be non-zero constants and they generate a ring that is isomorphic to the field of coefficients but the non-unit element $x$ cannot be generated by them.

Answer 2

Perhaps similar to what you are asking for, is the group ring construction. It is the left adjoint to the group of units functor. ("Adjunction" corresponds loosely to the notion of "equivalence" in category theory.)