RO-NMO 2010 Problem 122

We say that a ring $A$ has property $(P)$ if any non-zero element can be written uniquely as the sum of an invertible element and a non-invertible element. a) If in $A$ , $1+1=0$ , prove that $A$ has property $(P)$ if and only if $A$ is a field. b) Give an example of a ring that is not a field, containing at least two elements, and having property $(P)$ .

Dan Schwarz