Some remarks on symmetric linear functions and pseudotrace maps

Let A be a finite-dimensional associative algebra and $ϕ$ a symmetric linear function on $A$. In this note, we will show that the pseudotrace maps are obtained as special cases of well-known symmetric linear functions on the endomorphism rings of projective modules. We also prove that modules are interlocked with $ϕ$ if and only if they are projective. As an application of our approach, we will give proofs of several propositions and theorems for pseudotrace maps for an arbitrary finite-dimensional associative algebra.

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