Simplificator property of algebras with weak reduction

We construct an analogue of Gröbner bases for one-sided ideals in weakly reduced algebras. Diamond lemma is proved, it gives a modification of Buchberger algorithm for bases computation. We describe an explicite form of bases in principal ideal, in particular it is finite and we have an upper bound for degrees of bases elements. We compute the generators for modules of syzygies, as a corollary we give a simple test whether or not an element is zero divisor.