Reverse derivations with invertible values

الملخص EN

In this paper, we will prove the following theorem, Let R be a ring with 1 having a reverse derivation d ≠ 0 such that, for each x R, either d(x) = 0 or d(x) is invertible in R, then R must be one of the following: (i) a division ring D, (ii) D , the ring of 2×2 matrices over D, (iii) D[x]/(x where char D = 2, d (D) = 0 and d(x) = 1 + ax for some a in the center Z of D.

Furthermore, if 2R ≠ 0 then R = D is possible if and only if D does not contain all quadratic extensions of Z, the center of D.