Exercise 8, § 4

Let A be a left Noetherian ring.

a) Let $x$ be a nonzero element of A. Prove that the left annihilator Ann $x$ of $x$ is not essential (among the elements that do not have this property, take an $x$ such that Ann $x$ is maximal, and consider $Ax\cap$ Ann $x$).

b) Let $x$ be a right cancellable element of A. Prove that the ideal $Ax$ is essential (if $\mathfrak{a}$ is an ideal of A such that $\mathfrak{a}\cap Ax= 0$, consider the ideal $\mathfrak{a}+\mathfrak{a}x+\mathfrak{a}x^2+. . .$).

c) Deduce from a) and b) that every right cancellable element of A is cancellable. d) Let $\mathfrak{a}$ be an essential left ideal of A containing a nonnilpotent element. Prove that $\mathfrak{a}$ contains a cancellable element (let $(x_1, . . . , x_n)$ be a sequence of nonzero elements of $\mathfrak{a}$ satisfying Ann $x^2_i=$ Ann $x_i$ and $x_{i+1}\in$ Ann $x_i\cap \cdots \cap$ Ann $x_1$ for every $i\geqslant 1$; observe that $\mathfrak{a}$ contains the direct sum of the $Ax_i$. Deduce that there exists such a sequence with $\bigcap_i$ Ann $x_i= 0$; prove that the element $\sum x^2_i$ is then right cancellable, and use c)).