1. Preliminaries. For rings with identity, well known proper inclusions exist among the classes of semisimple, left hereditary, left semihereditary, left noetherian, yon Neumann regular, and left coherent rings. Now all but the first two of these classes involve certain notions of finiteness (for example, a ring is left noetherian if and only if all of its left ideals are finitely generated). In this paper, we shall investigate the above inclusions in the setting of rings with several objects, where the finiteness conditions have been replaced by corresponding conditions involving an arbitrary cardinal number Rp. For /~=>0, Rp denotes the/3th cardinal in the wellordered class of infinite cardinals and R-1 denotes any finite cardinal. We remark that, when used in classical finite settings, the R_ ~ either is replaced by "finitely" or does not occur. A ring with several objects is a small additive category, denoted by c~. Note that a ring with a single object is just a ring with identity. We ask that cg be small in order to guarantee that the class of covariant, additive functors from cg to d ~ can be realized as a category, denoted by -~r We remark that for a ring with identity R, the category d ~ R is (isomorphic to) the class of left R-modules RMod. A morphism r/E~c~(M, N) will be denoted by M3~.N. Also, for pElCgl=set of objects of c~, Cg(p, _ ) will denote the covariant horn fuactor from cg to ~r If {M~}a~A is a family of objects of d ~ ~e, we denote by ~M). the coproduct of the lira ~IAfamily. Finally, if {M~M,]2, pEA, )o~#} is a direct system in ~r then --X~,~ denotes the direct limit of the system. The following definitions are basic and will be used throughout this paper. In most cases a particular definition is a natural generalization of the classical definition for a ring with a sing!e object. In all cases, a statement gotten by setting ]~ = 1 is equivalent to the classical single object definition.