We know that every module $M$ is embedded in an injective module $D$. Is it true that the module $D/M$ is torsion?
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No, let $A = \frac{k[x, y]}{x^2, y^2}$ where $k$ is a field of characteristic $2$. This is a Frobenius algebra, hence it's self injective which means that $A$ is injective as a module over itself. As a module, the socle of $A$ is the principal ideal generated by $xy \in A$. This is $1$-dimensional so $A$ is indecomposable and hence the injective envelope of it's socle. The quotient, $\frac{k[x, y]}{(x, y)^2}$, of $A$ by its socle is not a torsion module. Remember: For an element $m \in M$ in a module to be torsion we must have $rm = 0$ for some nonzero $r \in A$ which is, additionally, not a zero-divisor. |
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