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This study introduces an ample Hausdorff groupoid Aˆ ⋊ R extracted from an ample Hausdorff groupoid G and a unital commutative ring R; a Hausdorff groupoid D which is the discrete twist over Aˆ ⋊ R. In the groupoid C*-algebra perspective, when R = C there is an isomorphism between the non-twisted groupoid C*-algebra (C∗(G)) and the twisted groupoid C*-algebra (C∗(Aˆ ⋊ R;D)). However, in this paper, in a purely algebraic setting, the non-twisted Steinberg algebra (AR(G)) and the twisted Steinberg algebra (AR(D;Aˆ⋊R)) are non-isomorphic. [ABSTRACT FROM AUTHOR]

In the present paper, we apply the concept of reverse derivation in rings on the concept of d - algebra to obtain the concept called a left-right (resp. right-left) reverse derivations of d-algebra X (briefly, (l, r) resp. (r, l)-reverse derivation of d-algebra), we will also, define some concepts such as regular map, composition two maps and study the related properties. Moreover, the notions of partial ordered edge d - algebra as well as d - subalgebra and their relation to our current study are obtained. In addition, some illustrative examples and counterexamples are discussed. [ABSTRACT FROM AUTHOR]