Solving the Word Problem for Two Classes of Non-associative Rings by Rewriting

Using the notion of modified completion given in Widiger, A. (1998, Deciding degree-four-identities for alternative rings by rewriting. In Bronstein, M., Grabmeier, J., Weispfenning, V. eds, Symbolic Rewriting Techniques, PCS 15, pp. 277–288. Birkhäuser-Verlag), it is shown that the word problem for the varieties of non-associative rings defined by (xy)z = y(zx) and (xy)z = y(xz) respectively can be decided by rewriting.