Question 180
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 $A$, $1 + 1 \neq 0$, 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).
a) If in $A$, $1 + 1 \neq 0$, 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).