If an R-module homomorphism vanishes on a submodule , there is a unique homomorphism satisfying , where is the quotient map. It is given by .
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In abstract algebra, the quotient module (also known as the factor module) is a construction that generalizes the notion of quotient spaces in linear algebra and topology. It is used in the context of modules over a ring, similar to how quotient groups are formed in group theory. ### Definition Let \( M \) be a module over a ring \( R \), and let \( N \) be a submodule of \( M \).