Algorithmic problems in associative algebras
The author proves that the occurrence problem for free algebras in the variety of algebras obeying the identity \([x_1,x_2][x_3,x_4][x_5,x_6][x_7,x_8]=0\) is undecidable. The next result is that there is no algorithm determining whether or not a finite system of elements in a free associative algebra is algebraically dependent. And the last result is that there exists a f.g. associative algebra with undecidable word problem, such that its relation module is free. These undecidability results are proven by interpretation of two-tape Minsky machines. In the proof of the second result the cycles in these machines are studied and interpreted as dependences in algebras.
scientific article; zbMATH DE number 738958