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