ABSTRACT
IV.1.1 All gradations considered in this chapter are Z-gradations. Recall that in this case a (commutative) graded field is of the form k or k[T,T −1], where k is a field and T a variable of degree t ∈ N0. An arbitrary graded field is of the form D or D[X,X−1,ψ] where D is a skewfield, ψ an automorphism of D, and X a variable of positive degree satisfying the commutation rule Xα = α φ X for any a ∈ D. Let R be a graded ring and d ∈ Z n. As before, we denote by R n (d) the graded freeR-module with homogeneous generators ex, e2,…,en, where degei = di . We let Mn(R)(d) = ENDR (Rn(d)). Then Mn(R)(d) is in fact the matrix ring Mn(R) with gradation given by Mn(R)(d) = https://www.w3.org/1998/Math/MathML"> { A ∈ M n ( R ) : deg A i j = α + d j − d i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003066040/0d08f023-3dcb-4de9-a766-20c465754fff/content/ieq0668.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> .
