• 대한수학회논문집
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• 제32권3호
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• pp.535-542
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• 2017

Let R be an associative ring and ${\alpha},{\beta}:R{\rightarrow}R$ ring homomorphisms. An additive mapping $d:R{\rightarrow}R$ is called an (${\alpha},{\beta}$)-derivation of R if $d(xy)=d(x){\alpha}(y)+{\beta}(x)d(y)$ is fulfilled for any $x,y{\in}R$, and an additive mapping $D:R{\rightarrow}R$ is called a generalized (${\alpha},{\beta}$)-derivation of R associated with an (${\alpha},{\beta}$)-derivation d if $D(xy)=D(x){\alpha}(y)+{\beta}(x)d(y)$ is fulfilled for all $x,y{\in}R$. In this note, we intend to generalize a theorem of Vukman [5], and a theorem of Daif and El-Sayiad [2].

• 대한수학회논문집
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• 제33권1호
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• pp.73-83
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• 2018

Let R be an associative ring with involution * and ${\alpha},{\beta}:R{\rightarrow}R$ ring homomorphisms. An additive mapping $d:R{\rightarrow}R$ is called an $({\alpha},{\beta})^*$-derivation of R if $d(xy)=d(x){\alpha}(y^*)+{\beta}(x)d(y)$ is fulfilled for any $x,y{\in}R$, and an additive mapping $F:R{\rightarrow}R$ is called a generalized $({\alpha},{\beta})^*$-derivation of R associated with an $({\alpha},{\beta})^*$-derivation d if $F(xy)=F(x){\alpha}(y^*)+{\beta}(x)d(y)$ is fulfilled for all $x,y{\in}R$. In this note, we intend to generalize a theorem of Vukman [12], and a theorem of Daif and El-Sayiad [6], moreover, we generalize a theorem of Ali et al. [4] and a theorem of Huang and Koc [9] related to generalized Jordan triple $({\alpha},{\beta})^*$-derivations.

• 대한수학회보
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• 제59권6호
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• pp.1387-1408
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• 2022

Let R be a commutative ring with a non-zero identity, S be a multiplicatively closed subset of R and M be a unital R-module. In this paper, we define a submodule N of M with (N :_R M)∩S = ∅ to be weakly S-prime if there exists s ∈ S such that whenever a ∈ R and m ∈ M with 0 ≠ am ∈ N, then either sa ∈ (N :_R M) or sm ∈ N. Many properties, examples and characterizations of weakly S-prime submodules are introduced, especially in multiplication modules. Moreover, we investigate the behavior of this structure under module homomorphisms, localizations, quotient modules, cartesian product and idealizations. Finally, we define two kinds of submodules of the amalgamation module along an ideal and investigate conditions under which they are weakly S-prime.

• 대한수학회논문집
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• 제38권1호
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• pp.21-38
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• 2023

Let R be a commutative ring with identity. Let R be an integral domain and M a torsion-free R-module. We investigate the relation between the notion of e-exactness, recently introduced by Akray and Zebari [1], and generalized the concept of homology, and establish a relation between e-exact sequences and homology of modules. We modify some applications of e-exact sequences in homology and reprove some results of homology with e-exact sequences such as horseshoe lemma, long exact sequences, connecting homomorphisms and etc. Next, we generalize two special drived functor T or and Ext, and study some properties of them.