Let \(\Lambda\) be a unital associative algebra over an algebraically closed field \(k\). Two relations are defined on finitely generated (f.g.) left \(\Lambda\)-modules: \(M\leq_{\hom}N\) iff \(\dim(X,M)\leq\dim(X,N)\) for all f.g. left \(\Lambda\)-modules; \(M\leq_{\text{stab}}N\) iff \(\dim\underline{(X,M)}\leq\dim\underline{(X,N)}\) for all f.g. left \(\Lambda\)-modules. Here \((X,Y)\) denotes the \(k\)-space of \(\Lambda\)-homomorphisms, and \(\underline{(X,Y)}\) is the \(k\)-space of \(\Lambda\)-stable homomorphisms.
The main results of the paper are: Let \(M,N\) be f.g. left \(\Lambda\)-modules with the same dimension vector. Then \(M\leq_{\hom}N\) implies \(M\leq_{\text{stab}}N\) (Proposition 1); The converse is true if \(\Lambda\) is hereditary (Proposition 2); There exists \(\Lambda\) such that the converse is not true (Proposition 3).