𝔏-Rickart Modules

It is well known that the Rickart property of rings is not a left-right symmetric property. We extend the notion of the left Rickart property of rings to a general module theoretic setting and define 𝔏-Rickart modules. We study this notion for a right R-module M _R where R is any ring and obtain its basic properties. While it is known that the endomorphism ring of a Rickart module is a right Rickart ring, we show that the endomorphism ring of an 𝔏-Rickart module is not a left Rickart ring in general. If M _R is a finitely generated 𝔏-Rickart module, we prove that End _R (M) is a left Rickart ring. We prove that an 𝔏-Rickart module with no set of infinitely many nonzero orthogonal idempotents in its endomorphism ring is a Baer module. 𝔏-Rickart modules are shown to satisfy a certain kind of nonsingularity which we term “endo-nonsingularity.” Among other results, we prove that M is endo-nonsingular and End _R (M) is a left extending ring iff M is a Baer module and End _R (M) is left cononsingular.     

Rickart Modules

The concept of right Rickart rings (or right p.p. rings) has been extensively studied in the literature. In this article, we study the notion of Rickart modules in the general module theoretic setting by utilizing the endomorphism ring of a module. We provide several characterizations of Rickart modules and study their properties. It is shown that the class of rings R for which every right R-module is Rickart is precisely that of semisimple artinian rings, while the class of rings R for which every free R-module is Rickart is precisely that of right hereditary rings. Connections between a Rickart module and its endomorphism ring are studied. A characterization of precisely when the endomorphism ring of a Rickart module will be a right Rickart ring is provided. We prove that a Rickart module with no infinite set of nonzero orthogonal idempotents in its endomorphism ring is precisely a Baer module. We show that a finitely generated module over a principal ideal domain (PID) is Rickart exactly if it is either semisimple or torsion-free. Examples which delineate the concepts and results are provided.     

On Weak Dual Rickart Modules and Dual Baer Modules

We introduce and study the notion of wd-Rickart modules (i.e. modules M such that for every nonzero endomorphism ? of M, the image of ? contains a nonzero direct summand of M). We show that the class of rings R for which every right R-module is wd-Rickart is exactly that of right semi-artinian right V-rings. We prove that a module M is dual Baer if and only if M is wd-Rickart and M has the strong summand sum property. Several structure results for some classes of wd-Rickart modules and dual Baer modules are provided. Some relevant counterexamples are indicated.     

t-Extending Modules and t-Baer Modules

We introduce the notions of “t-extending modules,” and “t-Baer modules,” which are generalizations of extending modules. The second notion is also a generalization of nonsingular Baer modules. We show that a homomorphic image (hence a direct summand) of a t-extending module and a direct summand of a t-Baer module inherits the property. It is shown that a module M is t-extending if and only if M is t-Baer and t-cononsingular. The rings for which every free right module is t-extending are called right Σ-t-extending. The class of right Σ-t-extending rings properly contains the class of right Σ-extending rings. Among other equivalent conditions for such rings, it is shown that a ring R is right Σ-t-extending, if and only if, every right R-module is t-extending, if and only if, every right R-module is t-Baer, if and only if, every nonsingular right R-module is projective. Moreover, it is proved that for a ring R, every free right R-module is t-Baer if and only if Z ₂(R _R ) is a direct summand of R and every submodule of a direct product of nonsingular projective R-modules is projective.     

Noetherian rings with almost injective simple modules

We characterize right Noetherian rings over which all simple modules are almost injective. It is proved that R is such a ring, if and only if, the complements of semisimple submodules of every R-module M are direct summands of M, if and only if, R is a finite direct sum of right ideals I_r, where I_r is either a Noetherian V-module with zero socle, or a simple module, or an injective module of length 2. A commutative Noetherian ring for which all simple modules are almost injective is precisely a finite direct product of rings R_i, where R_i is either a field or a quasi-Frobenius ring of length 2. We show that for commutative rings whose all simple modules are almost injective, the properties of Kasch, (semi)perfect, semilocal, quasi-Frobenius, Artinian, and Noetherian coincide.     

Rings all of whose right ideals are U-modules

The notion of a U-module was introduced and thoroughly investigated in [11 Ibrahim, Y., Yousif, M. F. (2017). U-modules. Comm. Algebra, doi:https://doi.org/10.1080/00927872.2017.1339064.[Crossref] , [Google Scholar]] as a strict and simultaneous generalization of quasi-continuous, square-free and automorphism-invariant modules. In this paper a right R-module M is called a U*-module if every submodule of M is a U-module, and a ring R is called a right U*-ring if R_R is a U*-module. We show that M is a U*-module iff whenever A and B are submodules of M with A?B and A∩B = 0, A⊕B is a semisimple summand of M; equivalently M = X⊕Y, where X is semisimple, Y is square-free, and X &; Y are orthogonal. In particular, a ring R is a right U*-ring iff R is a direct product of a square-full semisimple artinian ring and a right square-free ring. Moreover, right U*-rings are shown to be directly-finite, and if the ring is also an exchange ring then it satisfies the substitution property, has stable-range 1, and hence is stably-finite. These results are non-trivial extensions of similar ones on rings all of whose right ideals are either quasi-continuous or auto-invariant.