Romanian Mathematical Olympiad

a) If $A$ is a field and $x\in A$, $x\ne 0$, then $x$ is invertible and $x=x+0$; this representation is unique, since $0$ is the only noninvertible element.

Assume $A$ is not a field. Let $x\in A$, $x\ne 0$, be a noninvertible element. Since $1+x=(1+x)+0$ and $-1+x=(-1+x)+0$, the elements $1+x$ and $-1+x$ are noninvertible, otherwise the uniqueness of the representation would imply $x=0$.

Since $x=1+(-1+x)=-1+(1+x)$, it follows $1=-1$, contradiction.

b) Let $E$ be a nonempty set, and $A=(\mathcal{P}(E),\Delta ,\cap )$ be the Boolean ring of the set of parts of $E$. In this ring, $E$ is the only invertible element.

If $X$ is a nonempty part of $E$, then the element $E\setminus X$ is noninvertible, and $X=E\Delta (E\setminus X)$. The representation is unique, since $E$ is the only invertible element.