finiteness of a non-commutative ring

[xixi]

Let R be a non-commutative ring . Suppose that the number of non-units of R is finite . Can we say that R is a finite ring?

[zhentil]

If R and F are rings, then RxF is a ring. In particular, if R is a finite non-commutative ring and F is a field...

[xixi]

If R and F are rings, then RxF is a ring. In particular, if R is a finite non-commutative ring and F is a field...

so then??...

[Landau]

then RxF is an infinite non-commutative ring...
so....

[xixi]

then RxF is an infinite non-commutative ring...
so....

Do you mean that RxF is a counter-example ? but RxF has infinite non-unit elements .

[zhentil]

Silly me! I should read the question.