9) Let \( R \) denote a ring with \( 1 \neq 0 \). Let \( M \) denote an \( R \)-module, and let \( N \) denote an \( R \)-submodule of \( M. \)
Define \( J = \{r \in R \; | \; ra = 0 \) for any \( a \in N \}. \) Prove that \( J \) is a 2-sided ideal in \( R. \)
Proof. Suppose \( j, k \in J. \) Then \( ja = ka = 0 \) for all \( a \in N, \) so \( (j + k)a = ja + ka = 0 + 0 = 0 \), showing that \( J \) is additively closed.
For any \( r \in R, a \in N, (rj)a = r(ja) = r \cdot 0 = 0, \) and \( (jr)a = j(ra) = j \cdot 0 = 0. \) \( ra \in N \) is key for the second calculation; that is because
\( N \) is a submodule of \( N \), but all of \( N \) is annihilated by \( J \). Therefore \( J \) is a 2-sided ideal of \( R. \)QED.
Note: This is Ex. §10.1,#9 in Dummit & Foote.