Let \((S, {\mathfrak {n}})\) be a commutative noetherian local ring and let \(\omega \in {\mathfrak {n}}\) be non-zero divisor. This paper is concerned with the category of monomorphisms between finitely generated Gorenstein projective S-modules, such that their cokernels are annihilated by \(\omega \) . We will observe that this category, which will be denoted by \({\textsf{Mon}}(\omega , \mathcal {G} )\) , is an exact category in the sense of Quillen. More generally, it is proved that \({\textsf{Mon}}(\omega , \mathcal {G} )\) is a Frobenius category. Surprisingly, it is shown that not only the category of matrix factorizations embeds into \({\textsf{Mon}}(\omega , \mathcal {G} )\) , but also its stable category as well as the singularity category of the factor ring \(R=S/({\omega )}\) , can be realized as triangulated subcategories of the stable category \(\underline{{\textsf{Mon}}}(\omega , \mathcal {G} )\) .
