Suppose I have an category whose objects are free $R$-modules (R a polynomial ring) and whose morphism-spaces $\mathrm{Hom}(A,B)$ between objects $A$ and $B$ are spanned by a finite set of module-maps. A natural question to ask is if two objects in this category are isomorphic and, if they are, if I can find such an isomorphism explicitly. This boils down to finding $v \in \mathrm{Hom}(A,B)$ and $w \in \mathrm{Hom}(B,A)$ such that $$ v \circ w = \mathrm{Id}_B, \quad w \circ v = \mathrm{Id}_A. $$ Because the composition is $R$-bilinear it naturally extends to an $R$-linear map $$ \circ : \mathrm{Hom}(A,B) \otimes \mathrm{Hom}(B,A) \to \mathrm{Hom}(A,A). $$ If I use generating sets $\{f_i\}$ and $\{g_i\}$ for the $\mathrm{Hom}$-spaces I can construct an associated generating set $\{f_i\otimes g_j\}$ on the tensor product and the problem can be partially solved as a matrix-vector equation $[\circ] \cdot x = \mathrm{Id}_A$. However, the solutions to this equation are now tensors of the form $$ x_0 \cdot f_0 \otimes g_0 + \ldots + x_n \cdot f_n \otimes g_n. $$ If the maps are suitably perpendicular, in the sense that $f_i \circ g_j = 0$ for all $i\neq j$, then this can be written as a pure tensor $$ f\otimes g = (x_0 \cdot f_0 + \ldots + x_n \cdot f_n) \otimes (g_0 + \ldots + g_n), $$ and hopefully I find a that $f$ is an isomorphism. But in general this is not the case.
So basically what I want to ask is if there exists some sort of algorithm for solving the last step of this construction. In other words: how do I know if my tensor is pure?
