Introduction: Some Motivating Examples

The main difficulty in the studying the finitely presented noncommutative algebras is that in general most of the problems are unsolvable. For an illustration, consider the following example.

Example 1(Tsejtin, 1958) Let

A = 〈a, b, c, d|ac = ca, ad = da, bc = cb, bd = db, eca = ce, edb = de, cca = ccae〉.

It is impossible to give an algorithm determining whether two given words in the alphabet a, b, c, d, e are equal in the algebra A.

Nevertheless rather often (e.g. in the commutative or finite-dimensional cases) we are able to solve the corresponding word problem. The reason for this is the regular behaviour of the Gröbner bases in those cases. The aim of this article is to describe how to construct, analyse and use Gröbner bases in noncommutative algebras.

Let A = 〈X|R〉 be a finitely presented associative algebra over the field K. If I is the ideal generated by R in the free algebra K〈X〉 then A is isomorphic to the factor-algebra K〈X〉/I and every element of A can be presented as a linear combination of words in the alphabet X. Example 1 shows that one of the most important questions – which linear combinations present the same element in A – is far from trivial.