Exercise 20, § 1

Prove that a left Noetherian ring A without zero divisors admits a field of left fractions (Exercise 18) (if $a$ and $s$ are nonzero elements of A, use the fact that the sequence of ideals $(As+ Asa+\cdots + Asa^n)_{n\in\mathbf{N}}$ is stationary to prove that $Aa\cap As$ is not reduced to 0).