1.6.17 Fields, Groups, Rings

Problem: Let \( e_{1}, \ldots, e_{n} \) be such elements of the center of the ring \( A \) with such 1 , that \( 1=e_{1}+\cdots+e_{n}, e_{i}^{2}=e_{i} \), \( e_{i} e_{j}=0, i \neq j \). Prove that \( A e_{i} \) are two-sided ideals of the ring \( A \) and \( A=A e_{1} \oplus \ldots \oplus A e_{n} \).