What is a Trace Module? A Short Peek into Ring Theory

The sum of all $R$-homomorphisms from a module $M$ to $R$ is called the trace ideal of $M$ in $R$. The trace ideal is an object studied by ring theorists in a variety of contexts. We may utilize trace ideals, and by extension trace modules, to understand ring classification and module classifications. We will introduce the idea of a trace module which is the sum of all $R$-homomorphisms from a module $M$ to another module $N$. This is a sort of generalization of the trace ideal as it removes the restriction that the target module is $R$. The trace module may be used to explore the same two lines of questioning that the trace ideal is being used for