$\DeclareMathOperator\Tr{Tr}$Say that we have an algebra $R$ over $\mathbb{C}$. If, for two finitely generated $R$-modules $M, N$ we know that $\Tr_M(r)=\Tr_N(r)$ for all $r\in R$, where we consider $r$ as the linear endomorphism of the corresponding module, then we know that $M\cong N$ by the Brauer–Nesbitt theorem.

My question is, assuming that we have specific values for traces (some homomorphism $g: R \rightarrow \mathbb{C}$ with $g(rr')=g(r'r)$ for every $r, r'\in R$), do we know that there exists some module $M$ with $Tr_M(r)=g(r)$ for every $r$? If not, can we somehow efficiently describe the functions that can appear as traces?

$\DeclareMathOperator\Tr{Tr}$Say that we have an algebra $R$ over $\mathbb{C}$. If, for two finitely generated (edit: and semisimple) $R$-modules $M, N$ we know that $\Tr_M(r)=\Tr_N(r)$ for all $r\in R$, where we consider $r$ as the linear endomorphism of the corresponding module, then we know that $M\cong N$ by the Brauer–Nesbitt theorem.

My question is, assuming that we have specific values for traces (some homomorphism $g: R \rightarrow \mathbb{C}$ with $g(rr')=g(r'r)$ for every $r, r'\in R$), do we know that there exists some module $M$ with $Tr_M(r)=g(r)$ for every $r$? If not, can we somehow efficiently describe the functions that can appear as traces?

Say that we have an algebra $R$ over $\mathbb{C}$. If for two finitely generated $R-$modules $M, N$ we know that $Tr_M(r)=Tr_N(r)$ for all $r\in R$, where we consider $r$ as the linear endomorphism of the corresponding module, then we know that $M\cong N$ by the Brauer-Nesbitt theorem.

My question is, assuming that we have specific values for traces (some homomorphism $g: R \rightarrow \mathbb{C}$ with $g(rr')=g(r'r)$ for every $r, r'\in R$), do we know that there exists some module $M$ with $Tr_M(r)=g(r)$ for every $r$? If not, can we somehow efficiently describe the functions that can appear as traces?

$\DeclareMathOperator\Tr{Tr}$Say that we have an algebra $R$ over $\mathbb{C}$. If, for two finitely generated $R$-modules $M, N$ we know that $\Tr_M(r)=\Tr_N(r)$ for all $r\in R$, where we consider $r$ as the linear endomorphism of the corresponding module, then we know that $M\cong N$ by the Brauer–Nesbitt theorem.

My question is, assuming that we have specific values for traces (some homomorphism $g: R \rightarrow \mathbb{C}$ with $g(rr')=g(r'r)$ for every $r, r'\in R$), do we know that there exists some module $M$ with $Tr_M(r)=g(r)$ for every $r$? If not, can we somehow efficiently describe the functions that can appear as traces?