An R module M is herein called torsion if each element has nonzero annihilator, and faithful if the annihilator of M is zero. The central theme of this dissertation is exploration of which rings admit modules that are simultaneously faithful and torsion, termed FT modules. If a ring R admits an FT right module, it is called right faithful torsion or a right FT ring, and similarly for the left-hand side. The ring is said to have FT rank equal to κ if κ is a nonzero cardinal and is the least cardinality of a generating set for an FT module over R. By convention, rings which are not FT have FT rank 0.
After a survey of the requisite definitions from abstract algebra, several observations are made and lemmas are proven. It is shown that a ring with infinite right FT rank must have a properly descending chain of nonzero ideals of the same length as its FT rank. Using families of ideals with the finite intersection property, we construct torsion modules which are faithful when the family has intersection zero. Using this, it is possible to show that infinite FT ranks can only be regular cardinals. We determine the propagation of FT rank in standard ring constructions such as direct products, matrix rings, and the maximal right ring of quotients. To paraphrase the results: a product is FT if it has an FT factor, infinite products are always FT, matrix rings are often FT, and the FT property travels down from the maximal right ring of quotients.
The next portion of the dissertation gives an account of all that is known about several classes of rings and whether they are FT or not. The two prominent examples of rings that are not right FT are 1) quasi-Frobenius rings R such that R/rad(R) is a finite product of division rings, and 2) any ring R with an essential minimal right ideal. We show that finite products of simple rings are FT exactly when they are not finite products of division rings, with possible ranks 0 and 1. Domains are FT exactly when they are not division rings, with rank possible ranks 0, 1 or any infinite regular cardinal. We also show that quasi-Frobenius rings are FT exactly when R/rad(R) is not a finite product of division rings, with possible ranks 0 and 1. Right nonsingular rings are shown to be FT exactly when they are not finite products of division rings. Noetherian serial rings are shown to be FT exactly when they are not Artinian.
The next section demonstrates that for each possible infinite FT rank, there is a commutative domain with that FT rank. While many rings with FT rank 1 also exist, we note that it seems to be very hard to find an example of a ring with finite rank greater than 1.
In the next section, we reverse the central question by asking, "Given an abelian group, can the group can be made an FT module over some ring?" The main result is that a torsion abelian group G is FT over its endomorphism ring if and only if G is not cyclic.
Finally we explore a parallel definition of faithful singular modules and rings. We see that our definition of torsion generalizes that of singular modules, and that many of the questions we ask about FT rings are easily answered for faithful singular rings and modules. In particular the rings admitting faithful singular modules are completely classified: a ring R admits a faithful singular right module if and only if soc(RR)=0. Furthermore, a ring can have a faithful singular rank of any infinite regular cardinal, and also of 0 or 1, but not any other finite cardinal.