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Let $R$ be an arbitrary ring with identity and $M$ a right $R$-module with $S=$ End$_R(M)$. In this paper we introduce $pi$-Rickart modules as a generalization of generalized right principally projective rings as well as that of Rickart modules. The module $M$ is called {it $pi$-Rickart} if for any $fin S$, there exist $e^2=ein S$ and a positive integer $n$ such that $r_M(f^n)=eM$. We prove that several results of Rickart modules can be extended to $pi$-Rickart modules for this general settings, and investigate relations between a $pi$-Rickart module and its endomorphism ring.
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Let $R$ be an arbitrary ring with identity and $M$ a right $R$-module with $S=$ End$_R(M)$. Let $Z_2(M)$ be the second singular submodule of $M$. In this paper, we define Goldie Rickart modules by utilizing the endomorphisms of a module. The module $M$ is called Goldie Rickart if for any $fin S$, $f^{-1}(Z_2(M))$ is a direct summand of $M$. We provide several characterizations of Goldie Rickart modules and study their properties. Also we present that semisimple rings and right $Sigma$-$t$-extending rings admit some characterizations in terms of Goldie Rickart modules.
We initiate a study on a range of new generalized derivations of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero. This new generalization of derivations has an analogue in the theory of associative prime rings and unites many well-known generalized derivations that have already appeared extensively in the study of Lie algebras and other nonassociative algebras. After exploiting fundamental properties, we introduce and analyze their interiors, especially focusing on the rationality of the corresponding Hilbert series. Applying techniques in computational ideal theory we develop an approach to explicitly compute these new generalized derivations for the three-dimensional special linear Lie algebra over the complex field.
In this paper, we are interested in a class of modules partaking in the hierarchy of injective and cotorsion modules, so-called Harmanci injective modules, which turn out by the motivation of relations among the concepts of injectivity, flatness and cotorsionness. We give some characterizations and properties of this class of modules. It is shown that the class of all Harmanci injective modules is enveloping, and forms a perfect cotorsion theory with the class of modules whose character modules are Matlis injective. One of the main objectives we pursue is to know when the injective envelope of a ring as a module over itself is a flat module.
In this paper, we define the induced modules of Lie algebra ad$(B)$ associated with a 3-Lie algebra $B$-module, and study the relation between 3-Lie algebra $A_{omega}^{delta}$-modules and induced modules of inner derivation algebra ad$(A_{omega}^{delta})$. We construct two infinite dimensional intermediate series modules of 3-Lie algebra $A_{omega}^{delta}$, and two infinite dimensional modules $(V, psi_{lambdamu})$ and $(V, phi_{mu})$ of the Lie algebra ad$(A_{omega}^{delta})$, and prove that only $(V, psi_{lambda0})$ and $(V, psi_{lambda1})$ are induced modules.
It is proved that if A_p is a countable elementary abelian p-group, then: (i) The ring End(A_p) does not admit a nondiscrete locally compact ring topology. (ii) Under (CH) the simple ring End(A_p)/I, where I is the ideal of End(A_p) consisting of all endomorphisms with finite images, does not admit a nondiscrete locally compact ring topology. (iii) The finite topology on End(A_p) is the only second metrizable ring topology on it. Moreover, a characterization of completely simple endomorphism rings of the endomorphism rings of modules over commutative rings is also obtained.
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