- #1

- 46

- 17

## Main Question or Discussion Point

Let ##M## be a (right) R-module, and ##A## and ##B## two submodules of ##M##.

If ##A = B ##, then we know that ##\frac{M}{A} = \frac{M}{B}##.

But is the converse also true:

If ##\frac{M}{A} = \frac{M}{B}##, can we conclude that ##A = B ## ?

I doubt it, but I cannot find the answer. Maybe someone can help me with a proof or a counterexample ?

(If possible, also a reference.)

If ##A = B ##, then we know that ##\frac{M}{A} = \frac{M}{B}##.

But is the converse also true:

If ##\frac{M}{A} = \frac{M}{B}##, can we conclude that ##A = B ## ?

I doubt it, but I cannot find the answer. Maybe someone can help me with a proof or a counterexample ?

(If possible, also a reference.)