Show that
$R$ is local if and only if for every choice of finite elements $r_1, ..., r_m \in R$ this implication applies: $$\sum \limits_{i=1}^{m}r_i \in R^* \implies\exists i \in\{1,...,m\}\text{ with }r_i \in R^*.$$
Now I already got: "$\Rightarrow$" and for "$\Leftarrow$" the case if there can only be one maximal ideal (if we suppose that there is more than one maximal ideal). But I don't know how to show the existance of a maximal ideal. Please give me a hint.
Every (commutative, unital) ring has at least one maximal ideal. You can show any proper ideal is contained in a maximal ideal using Zorn's lemma, and any non-unit of $R$ generates a proper ideal.