In this paper we show that a semi-commutative Galois extension of associative unital algebra by means of an element τ, which satisfies τ N = 1 (1 is the identity element of an algebra and N ≥ 2 is an integer) induces a structure of graded q-differential algebra, where q is a primitive Nth root of unity. A graded qdifferential algebra with differential d, which satisfies d N = 0, N ≥ 2, can be viewed as a generalization of graded differential algebra. The subalgebra of elements of degree zero and the subspace of elements of degree one of a graded q-differential algebra together with a differential d can be considered as a first order noncommutative differential calculus. In this paper we assume that we are given a semicommutative Galois extension of associative unital algebra, then we show how one can construct the graded q-differential algebra and when this algebra is constructed we study its first order noncommutative differential calculus. We also study the subspaces of graded q-differential algebra of degree greater than one which we call the higher order noncommutative differential calculus induced by a semi-commutative Galois extension of associative unital algebra. We also study the subspaces of graded q-differential algebra of degree greater than one which we call the higher order noncommutative differential calculus induced by a semi-commutative Galois extension of associative unital algebra. Finally we show that a reduced quantum plane can be viewed as a semi-commutative Galois extension of a fractional one-dimensional space and we apply the noncommutative differential calculus developed in the previous sections to a reduced quantum plane.
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