When do the units of a ring together with 0 form a field?

Question

Let $R$ be a commutative ring with identity. Let $U(R)$ be the multiplicative group of units of $R$. For what rings $R$ does $U(R)\cup\{0\}$ form a field? This is equivalent to asking: for what commutative rings is it true that the sum of two units is a unit or 0?

Fields trivially have this property. A nontrivial example is a polynomial ring over a field, since units look like $a_0+a_1x+\ldots+a_nx^n$ with $a_0$ a unit and $a_1,\ldots,a_n$ nilpotent. Edit: fields have no nontrivial nilpotent elements, so this example is also pretty trivial. I would appreciate any interesting examples of rings with the described property.

Relevant, but without conclusive answers: When does sum of two units give a unit?

Answer 1

There might not be a thorough classification of such rings.

You may give it a name, though, say a "good ring" is a ring in which the sum of two units is either $0$ or a unit.

It is then possible to deduce some properties and equivalent definitions of good rings.

For example (all the rings I talk about are commutative):

A ring $R$ is good if and only if it contains a subfield $k$ such that $R^\times = k^\times$.

A ring $R$ is good if and only if for any $1 \neq u\in R^\times$, one has $u - 1 \in R^\times$.

If $R$ is a good ring, then the polynomial ring $R[x]$ is also good.

Note that the last construction can be iterated, thus e.g. $k[x_1, \dotsc, x_n]$ is good.

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The realistic question is how useful this definition is. Usually, notions in commutative algebra serve as tools for other subjects such as algebraic geometry, algebraic number theory, etc.

That said, of course one can always study something just for fun, and may even find great applications later.