is actually more than just a subring ... it's an ideal of A! prove it! (J is called the annihilator of M in A) to show that M has a structure of an R/J module, we only need to show that the defined

action is basically well-defined, because other properties of modules will be directly transfered from R-module structure of M. so suppose then thus by the definition

of J, we have: hence: Q.E.D.