Some remarks on symmetric linear functions and pseudotrace maps

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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.