Adnan Tercan

Goldie Extending Modules

In this article, we define a module M to be 𝒢-extending if and only if for each X ≤ M there exists a direct summand D of M such that X ∩ D is essential in both X and D. We consider the decomposition theory for 𝒢-extending modules and give a characterization of the Abelian groups which are 𝒢-extending. In contrast to the charac-terization of extending Abelian groups, we obtain that all finitely generated Abelian groups are 𝒢-extending. We prove that a minimal cogenerator for 𝒢od-R is 𝒢-extending, but not, in general, extending. It is also shown that if M is (𝒢-) extending, then so is its rational hull. Examples are provided to illustrate and delimit the theory.

When Some Complement of a Submodule Is a Summand

A module M is said to satisfy the C 11 condition if every submodule of M has a (i.e., at least one) complement which is a direct summand. It is known that the C 1 condition implies the C 11 condition and that the class of C 11-modules is closed under direct sums but not under direct summands. We show that if M = M 1 ⊕ M 2, where M has C 11 and M 1 is a fully invariant submodule of M, then both M 1 and M 2 are C 11-modules. Moreover, the C 11 condition is shown to be closed under formation of the ring of column finite matrices of size Γ, the ring of m-by-m upper triangular matrices and right essential overrings. For a module M, we also show that all essential extensions of M satisfying C 11 are essential extensions of C 11-modules constructed from M and certain subsets of idempotent elements of the ring of endomorphisms of the injective hull of M. Finally, we prove that if M is a C 11-module, then so is its rational hull. Examples are provided to illustrate and delimit the theory.

The Extending Condition Relative to Sets of Submodules

A module M is called an extending (or CS) module provided that every submodule of M is essential in a direct summand of M. We call a module 𝒞-extending if every member of the set 𝒞 is essential in a direct summand where 𝒞 is a subset of the set of all submodules of M. Our focus is the behavior of the 𝒞-extending modules with respect to direct sums and direct summands. By obtaining various well-known results on extending modules and generalizations as corollaries of our results, we show that the 𝒞-extending concept provides a unifying framework for many generalizations of the extending notion. Moreover, by applying our results to various sets 𝒞, including the projection invariant submodules, the projective submodules, and torsion or torsion-free submodules of a module, we obtain new results including a characterization of the projection invariant extending Abelian groups.

The conditions (Ci) in modular lattices, and applications

In this paper, we introduce and investigate the latticial counterparts of the conditions (Ci), i = 1, 2, 3, 11, 12, for modules. In particular, we study the lattices satisfying the condition (C1), we call CC lattices (for Closed are Complements), i.e. the lattices such that any closed element is a complement, that are the latticial counterparts of CS modules (for Closed are Summands). Applications of these results are given to Grothendieck categories and module categories equipped with a torsion theory.

Modules Whose Submodules are Essentially Embedded in Direct Summands

A module M is said to satisfy the C 12 condition if every submodule of M is essentially embedded in a direct summand of M. It is known that the C 11 (and hence also C 1) condition implies the C 12 condition. We show that the class of C 12-modules is closed under direct sums and also essential extensions whenever any module in the class is relative injective with respect to its essential extensions. We prove that if M is a -module with cancellable socle and satisfies ascending chain (respectively, descending chain) condition on essential submodules, then M is a direct sum of a semisimple and a Noetherian (respectively, Artinian) submodules. Moreover, a C 12-module with cancellable socle is shown to be a direct sum of a module with essential socle and a module with zero socle. An example is constructed to show that the reverse of the last result do not hold.

Goldie Extending Rings

In this article, we investigate the 𝒢-extending condition under various ring extensions. We show that if R R is 𝒢-extending and S is a right essential overring, then S R and S S are 𝒢-extending. For split-null extensions, we show that if M ⊴ R and M is left faithful, then R R is (𝒢-) extending if and only if S S is (𝒢-) extending, where S = S(R, M). This result appears to be new for the extending case. We conclude with results on Dorroh extensions.

Projection invariant extending rings

A ring R is said to be right π-extending if every projection invariant right ideal of R is essential in a direct summand of R. In this article, we investigate the transfer of the π-extending condition between a ring R and its various ring extensions. More specifically, we characterize the right π-extending generalized triangular matrix rings; and we show that if RR is π-extending, then so is TT where T is an overring of R which is an essential extension of R, an n × n upper triangular matrix ring of R, a column finite or column and row finite matrix ring over R, or a certain type of trivial extension of R.

Goldie-Rad-Supplemented Modules

Abstract In this paper we introduce β** relation on the lattice of submodules of a module M. We say that submodules X, Y of M are β** equivalent, X β** Y, if and only if X+YX⊆Rad(M)+XX

Ring and Module Theory

{{X + Y} \over X} \subseteq {{Rad(M) + X} \over X}

Corrigendum To: Goldie Extending Modules

and X+YY⊆Rad(M)+YY