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Since this question Characteristics of a pretty ring I am interested in "pretty rings".

Definition: A pretty ring $R$ is a ring with unity 1, not a field, and each nonzero element can be written uniquely as a sum of a unit and a nonunit element of $R$.

I've already asked a question about the structure of pretty rings (namely here Characterization of pretty rings) and realized that my intuition about their structure was not quite right. For instance, we can have noncommutative pretty rings. For example the polynomial ring in two noncommutative variables over $\mathbb{F}_2$ is pretty and noncommutative.

1.) Does there exist a finite pretty ring nonisomorphic to $\mathbb{F}_2^{\oplus n}$?

and can we go further and remove commutativity in the finite case?

2.) Does there exist a finite noncommutative pretty ring?

I tried to construct a counterexample myself picking a noncommutative monoid and considering $R_S:=\mathbb{F}_2[S]$. This is a noncommutative unital ring (also finite if $S$ is finite). However, I fail to see how to pick the monoid such that $\vert R_S^\times \vert =1$ (which is equivalent to being a pretty ring, see my answer to this question Characteristics of a pretty ring).

A necessary condition is that $S$ has only one invertible element, but this is not sufficient. Namely, consider $S=\{ 1, a, b\}$ with $1$ being the neutral element and for $x,y \in \{ a, b\}$ we define $$ x\cdot y := x.$$ Then $R_S$ is noncommutative and unital, but $$ (1 + a + b)^2 = 1 + 4 a + 4 b =1 $$ and thus $R_S$ is not pretty.

I am aware that that not all pretty rings are isomorphic to some $R_S$ (for example the polynomial ring is not), however, I couldn't come up with a better way of thinking about this problem.

3.) Do the pretty rings have a "real" name? Are there people studying them?

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The answer to 1) is Yes and 2) is No.

Earlier you established they have exactly one unit, and that implies they are reduced. A finite reduced ring is a finite direct product of fields (by the Artin-Wedderburn theorem and Wedderburn's little theorem).

Such a product must be commutative, so this answers 2).

Clearly if any of the fields is something more than $F_2$, you have extra units. So all the fields have to be isomorphic to $F_2$. This answers 1).

3) I've never heard of them. The uniqueness condition is highly restrictive. TBH I am not sure why one would choose this definition given that "has only one unit" is so much more concise.

A more popular subject of study are clean rings in which elements are the sum of a unit and idempotent. There is a notion of a uniquely clean ring which additionally requires uniqueness.


Update

Just now I ran across a paper defining a generalization of clean rings called left unit fusible rings which is similar to what you describe with uniqueness dropped, and "nonunit" replaced by "zero divisor."

Ghashghaei, E., & McGovern, W. W. (2017). Fusible rings. Communications in Algebra, 45(3), 1151-1165.

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  • $\begingroup$ Great! Very neat proof. I agree that the definition is nicer stated in the terms of having only one unit, however, I wanted to keep the one in the original question. Thanks for the paper. $\endgroup$ Jun 18 '19 at 21:06

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