The Ideal Membership Problem in Non-Commutative Polynomial Rings

LetXbe a non-commutative monoid with term order; letRbe a commutative, unital ring; letIbe an ideal in the non-commutative polynomial ringR X and let ƒ R X . In this setting the problem of determining whether ƒ Iis studied. In a manner analogous to the commutative case, see Möller (1989), weak Gröbner bases are defined and their basic properties are studied. We will see that in the non-commutative setting, when the coefficient ring is not a field, and when we enlarge the polynomial ring by adding more variables, weak Gröbner bases may exhibit unpleasant behavior that has no analog in the commutative case. Quite in general for ƒ R X , it is undecidable whether ƒ I. This follows from the fact that the word problem for free semigroups is undecidable. IfIis generated by a recursively enumerable set, then we give a semi-decision procedure that halts if and only if ƒ I. Finally we examine a class of nicely behaved ideals for which weak Gröbner bases can be easly computed.