The word problem and the ideal membership problem

This dissertation concerns two problems from computational algebra, the word problem for semigroups and the ideal membership problem for noncommutative polynomial rings. Historically, the word problem provided one of the first examples of an algorithmically unsolvable problem from outside of logic and computability theory. In terms of solvability, the word problem is equivalent to a restricted version of the membership problem. For ideals whose membership problem is solvable, computational techniques such as Grobner basis methods often can be used to solve the problem, but not always. In Chapter two, we develop a method which can be used to solve the membership problem for every ideal of a certain class whose membership problem is solvable. In addition, we obtain useful characterizations of those semigroups having a solvable word problem and certain finitely generated ideals having a solvable membership problem. This characterization is extended to a larger class of ideals in Chapter three. Finally, we investigate the word problem for one-relator semigroups in Chapter four. In particular, we show that if there is a one-relator semigroup having an unsolvable word problem, then there is such a semigroup M satisfying (1) the defining relation for M must satisfy certain restrictions concerning the number of occurrences of each generator, and (2) the problem of determining whether or not two words having the same number of occurrences of each generator are equivalent in M is not solvable.