Noncommutative matrix factorizations with an application to skew exterior algebras

Theory of matrix factorizations is useful to study hypersurfaces in commutative algebra. To noncommutative hypersurfaces, which are important objects algebraic geometry, we introduce a notion factorization for an arbitrary nonzero non-unit element ring. First show that the category graded invariant under operation called twist (this result generalization by Cassidy-Conner-Kirkman-Moore). Then give two equivalences involving and totally reflexive modules analogous famous Eisenbud hypersurfaces). As application, describe indecomposable over skew exterior algebras.