R. Then the direct sums and are isomorphic R-modules if and only if n = t and there exist two permutations and of 1, 2, ..., n such that and for every i = 1, 2, ..., n. This result, due to Facchini, has been extended to infinite direct sums of uniserial modules by Příhoda in 2006. This extension involves the so-called quasismall uniserial modules. These modules were defined by Nguyen Viet Dung and Facchini, and their existence was proved by Puninski. The weak form of the Krull-Schmidt Theorem holds not only for uniserial modules, but also for several other classes of modules (biuniform modules, cyclically presented modules over serial rings, kernels of morphisms between indecomposable injective modules, couniformly presented modules.) Notes on alternate, similar and related terms Right uniserial rings can also be referred to as right chain rings or right valuation rings. This latter term alludes to valuation rings, which are by definition commutative, uniserial domains. By the same token, uniserial modules have been called chain modules, and serial modules semichain modules. The notion of a catenary ring has "chain" as its namesake, but it is in general not related to chain rings. In the 1930s, Gottfried Köthe and Keizo Asano introduced the term Einreihig (literally "one-series") during investigations of rings over which all modules are direct sums of cyclic submodules . For this reason, uniserial was used to mean "Artinian principal ideal ring" even as recently as the 1970s. Köthe's paper also required a uniserial ring to have a unique composition series, which not only forces the right and left ideals to be linearly ordered, but also requires that there be only finitely many ideals in the chains of left and right ideals. Because of this historical precedent, some authors include the Artinian condition or finite composition length condition in their definitions of uniserial modules and rings. Expanding on Köthe's work, Tadashi Nakayama used the term generalized uniserial ring to refer to an Artinian serial ring. Nakayama showed that all modules over such rings are serial. Artinian serial rings are sometimes called Nakayama algebras, and they have a well-developed module theory. Warfield used the term homogeneously serial module for a serial module with the additional property that for any two finitely generated submodules A and B, where J(−) denotes the Jacobson radical of the module . In a module with finite composition length, this has the effect