Title:

CSmodules and generalizations

As a generalization of the divisibility of an abelian group, injectivity was defined for modules by Baer in 1940. Since then this concept has attracted much interest. The starting point of this thesis is that for any torsionfree abelian group A (Zmodule) let B ≤ A such that A / B is torsionfree, can any homomorphism ϕ : B → A be lifted to A (i.e. does there exist a homomorphism theta : A → A such that theta  B = ϕ)? Since the answer is no, it is decided to investigate lifting homomorphisms from submodules to M and relationships with one (or two) of the following properties: (C1) Every submodule of M is essential in a direct summand of M. Equivalently, every complement submodule of M is a direct summand of M. (C2) Every submodule isomorphic to a direct summand of M is itself a direct summand of M. (C ) If M1 and M2 are direct summands of M with M1 ∩ M2 = 0 then M1 ⊕ M2 is a direct summand of M. A module with the property (C1) is called a CSModule and a CSmodule with the property (C2) ((C3)) is called continuous (quasicontinuous) module. In particular Kamal and Muller's result: "MR satisfies (C1) if and only if M = Z2(M) ⊕ N and Z2(M) is Ninjective", allows us to consider nonsingular modules. Special rings are then considered and it is investigated when they are CSrings for nonsingular cases. In particular, let where S, T are rings and SMT bimodule such that SM is faithful. Then the necessary and sufficient conditions for R to be a right nonsingular right CSring are given. In general, the full matrix ring over a CSring needs not be a CSring. This thesis contains the equivalent conditions for a full matrix ring being CS over a domain. Kamal and Muller proved that over a commutative integral domain, any torsionfree reduced CSmodule is a finite direct sum of uniform modules. This result is generalized to nonsingular modules over a commutative ring with finitely many minimal prime ideals. This thesis also deals with the characterization of continuous and quasicontinuous modules in terms of lifting homomorphisms. Since the direct sum of two CSmodules need not be a CSmodule (C1) is weakened to (C11) as follows: (C11) Every submodule of M has a complement which is a direct summand of M. In contrast to CSmodules it is observed that any direct sum of modules with (C11) satisfies (C11). However, it is not possible to determine whether any direct summand of a module with (C11) satisfies (C11) or not. A module M satisfies (C11+) if any direct summand of M satisfies (C11). Moreover the weaker condition than (C11) is given as follows: (C12) For every submodule N of M there exists a direct summand K of M and a monomorphism a : N 4 K such that a(N) is essential in K. It is worth knowing whether any direct summand of M is a direct sum of uniform modules whenever M is itself a direct sum uniform modules. It was shown that this is true for modules over Z. The work was completed by considering conditions on a module M which imply that M is a direct sum of uniform modules and chain conditions with (C11+).
