New proofs of some properties of essential submodules

In 1986, Lambek [Lambek, J., Lectures on rings and modules, Chelsea Publishing Company New York, 1986] studied the connections between essential submodules and quotient rings, and in 2009, Steinberg [Steinberg, S. A., Lattice-ordered rings and modules, Springer New York Dordrecht Heidelberg London, 2009, 1–307] presented some interesting results in this field. The main aim of this paper is to provide proofs for those of these results for which we did not found old ones in the literature we studied, using the same techniques as in the mentioned books