Originally Posted by

**Arczi1984** Hi,

Could You help me with this task?

"Proof that a commutative ring a is a local ring iff for any $\displaystyle a, b \in A$, from this that $\displaystyle a+b=1$ we have $\displaystyle a$ is an invertible element or $\displaystyle b$ is an invertible element."

**Proof.** $\displaystyle (\Rightarrow)$ $\displaystyle a,b\in A$ - any element which satysfied equality $\displaystyle a+b=1$ and suppose that $\displaystyle a,b\notin U(A)$, where $\displaystyle U(A)$ is a set of invertible elements of $\displaystyle A$. We have then that $\displaystyle a,b\in \mathfrak{m}$. Since $\displaystyle \mathfrak{m}$ is an ideal, then also $\displaystyle 1=a+b\in \mathfrak{m}$, and so $\displaystyle \mathfrak{m}=A$. This is contradiction, so $\displaystyle a \in U(A)$ or $\displaystyle b \in U(A)$.

$\displaystyle (\Leftarrow)$ - I'm not sure. Could You help me with this implication?