  In recent work of T. Cassidy and the author, a notion of complete
intersection was defined for (non-commutative) regular skew polynomial rings,
defining it using both algebraic and geometric tools, where the commutative
definition is a special case. In this article, we extend the definition to a
larger class of algebras that contains regular graded skew Clifford algebras,
the coordinate ring of quantum matrices and homogenizations of universal
enveloping algebras. Regular algebras are often considered to be
non-commutative analogues of polynomial rings, so the results herein support
that viewpoint.
