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Motivated by recent result of P\'erez and R.G. on equality of test ideal ofmodule closure operation and trace ideal, and the well-known result by Smiththat parameter test ideal cannot be contained in parameter ideals, we study theobstruction of containment of trace ideals in ideals of finite projective (orinjective) dimension. As consequences of our results, we give upper bounds on$\mathfrak{m}$-adic order of trace ideals of certain modules over localCohen--Macaulay rings. We also prove analogous results for ideal of entries ofmaps in a free resolution of modules.

On containment of trace ideals in ideals of finite homological dimension