# Nonzero module with no nonzero finitely presented submodule

My question is: Does there exist a nonzero module over a non-Noetherian ring with no nonzero finitely presented submodule?

For any element $$m$$ of a left (right) $$R$$-module $$M$$, the submodule $$Rm$$ ($$mR$$) is finitely generated, but not finitely presented unless $$Ann(m)$$ is a finitely generated left (right) ideal of $$R$$.

Also, while a ring can have a non-finitely presented principal ideal (e.g. the principal ideal generated by $$x_1 + (x_1x_2, x_1x_3, x_1x_4, ...)$$ in $$\mathbb{Z}[(x_n)_{n \ge 1}]/(x_1x_2, x_1x_3, x_1x_4, ...)$$), any ring is obviously a finitely presented module over itself, so modules that answer the question must be different from the base ring.

Sure. For instance, let $$R$$ be any ring with a maximal (left) ideal $$I$$ which is not finitely generated, and let $$M=R/I$$. Then the only nonzero submodule of $$M$$ is $$M$$ itself, which is not finitely presented.