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In this presentation, we presents some definitions and the result of module theory which states that two injective modules are isomorphic when they are isomorphic to submodules of each other, which it is better known as Bumby's theorem [1]. Also we provide a generalization of it. More precisely, we show here that two RD-injective (respectively, pure-injective) modules over an integral domain are isomorphic if they are isomorphic to relatively divisible (respectively, pure) submodules of each other.

References:

1. R.T. Bumby, Modules which are isomorphic to submodules of each other. Arch Math. (Basel) 16 (1965), pp. 184–185)