On General Factorization Problems of n-D Polynomial Matrices

Different from previous perspective, this brief studies relationships between the modules $\rho ({\mathbf {F}}_{1})$ and $\rho ({\mathbf {F}})$ generated by the rows of ${\mathbf {F}}_{1}$ and ${\mathbf {F}}$ respectively under the assumption of ${\mathbf {F}}$ admitting a general factorization ${\mathbf {F}}={\mathbf {G}}_{1}{\mathbf {F}}_{1}$ . Two necessary and sufficient conditions for the equivalence of $\rho ({\mathbf {F}}_{1})$ and quotient modules of $\rho ({\mathbf {F}})$ with respect to some polynomials are obtained for two different situations. As a by-product, some general factorizations of ${\mathbf {F}}$ by computing related quotient modules can be got.