• Bulletin of the Korean Mathematical Society
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• v.55 no.5
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• pp.1333-1338
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• 2018

A ${\ast}-ring$ R is called ${\ast}-regular$ if every principal one-sided ideal of R is generated by a projection. In this note, several characterizations of ${\ast}-regular$ rings are provided. In particular, it is shown that a matrix ring $M_n(R)$ is ${\ast}-regular$ if and only if R is regular and $1+x^*_1x_1+{\cdots}+x^*_{n-1}x_{n-1}$ is a unit for all $x_i$ of R; which answers a question raised in the literature recently.

• Bulletin of the Korean Mathematical Society
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• v.58 no.4
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• pp.837-851
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• 2021

Let R ⊆ L ⊆ S be ring extensions. Two star operations ${\ast}_1{\in}Star(R,S)$, ${\ast}_2{\in}Star(L,S)$ are said to be linked if whenever $A^{{\ast}_1}= R^{{\ast}_1}$ for some finitely generated S-regular R-submodule A of S, then $(AL)^{{\ast}_2}=L^{{\ast}_2}$. We study properties of linked star operations; especially when ${\ast}_1$ and ${\ast}_2$ are strict star operations. We introduce the notion of Prüfer star multiplication extension ($P{\ast}ME$) and we show that under appropriate conditions, if the extension R ⊆ S is $P{\ast}_1ME$ and ${\ast}_1$ is linked to ${\ast}_2$, then L ⊆ S is $P{\ast}_2ME$.

• Kyungpook Mathematical Journal
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• v.58 no.3
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• pp.481-488
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• 2018

Let I be a ${\ast}$-ideal on a 2-torsion free ${\ast}$-prime ring and $F:R{\rightarrow}R$ a generalized derivation with an associated derivation $d:R{\rightarrow}R$. The aim of this paper is to explore the condition under which generalized derivation F becomes a left centralizer i.e., associated derivation d becomes a trivial map (i.e., zero map) on R.

• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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• pp.57-71
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• 2019

Let R be an associative ring with identity, M a t.u.p. monoid with only one unit and ${\omega}:M{\rightarrow}End(R)$ a monoid homomorphism. Let R be a reversible, M-compatible ring and ${\alpha}=a_1g_1+{\cdots}+a_ng_n$ a non-zero element in skew monoid ring $R{\ast}M$. It is proved that if there exists a non-zero element ${\beta}=b_1h_1+{\cdots}+b_mh_m$ in $R{\ast}M$ with ${\alpha}{\beta}=c$ is a constant, then there exist $1{\leq}i_0{\leq}n$, $1{\leq}j_0{\leq}m$ such that $g_{i_0}=e=h_{j_0}$ and $a_{i_0}b_{j_0}=c$ and there exist elements a, $0{\neq}r$ in R with ${\alpha}r=ca$. As a consequence, it is proved that ${\alpha}{\in}R*M$ is unit if and only if there exists $1{\leq}i_0{\leq}n$ such that $g_{i_0}=e$, $a_{i_0}$ is unit and aj is nilpotent for each $j{\neq}i_0$, where R is a reversible or right duo ring. Furthermore, we determine the relation between clean and nil clean elements of R and those elements in skew monoid ring $R{\ast}M$, where R is a reversible or right duo ring.

• Journal of the Chungcheong Mathematical Society
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• v.30 no.2
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• pp.251-258
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• 2017

The aim of the present paper is to establish some results involving generalized ${\ast}$-derivations in ${\ast}$-rings and investigate the commutativity of prime ${\ast}$-rings admitting generalized ${\ast}$-derivations of R satisfying certain identities and some related results have also been discussed.

• Journal of the Chungcheong Mathematical Society
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• v.30 no.3
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• pp.313-321
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• 2017

In this paper, we introduce the notion of a ${\ast}$-semiderivation on ${\ast}$-rings, and we try to extend some results for derivations of rings or near-rings to a more general case for ${\ast}$-semiderivations of prime ${\ast}$-rings.

• Bulletin of the Korean Mathematical Society
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• v.47 no.5
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• pp.907-913
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• 2010

Let D be an integrally closed domain with quotient field K, * be a star operation on D, X, Y be indeterminates over D, $N_*\;=\;\{f\;{\in}\;D[X]|\;(c_D(f))^*\;=\;D\}$ and $R\;=\;D[X]_{N_*}$. Let b be the b-operation on R, and let $*_c$ be the star operation on D defined by $I^{*_c}\;=\;(ID[X]_{N_*})^b\;{\cap}\;K$. Finally, let Kr(R, b) (resp., Kr(D, $*_c$)) be the Kronecker function ring of R (resp., D) with respect to Y (resp., X, Y). In this paper, we show that Kr(R, b) $\subseteq$ Kr(D, $*_c$) and Kr(R, b) is a kfr with respect to K(Y) and X in the notion of [2]. We also prove that Kr(R, b) = Kr(D, $*_c$) if and only if D is a $P{\ast}MD$. As a corollary, we have that if D is not a $P{\ast}MD$, then Kr(R, b) is an example of a kfr with respect to K(Y) and X but not a Kronecker function ring with respect to K(Y) and X.

• Bulletin of the Korean Chemical Society
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• v.25 no.12
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• pp.1911-1916
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• 2004

The conformational preference of tetrahomodioxacalix[4]arenes with three different para substituents on the phenolic ring has been investigated by using ab initio molecular orbital theory (RHF/6-31$G^{\ast}$) and density functional theory (B3LYP/6-31$G^{\ast}$). The stability order is predicted to be cone > C-1,2-alternate > partial cone > 1,3-alternate > COC-1,2-alternate. The distorted cone conformation is found to be most stable in a gas phase and the calculated results are in agreement with the reported $^1$H NMR and X-ray experimental observations. The substitution of methylene with dimethyleneoxa bridges increases the size of the annulus of the molecule, its conformational mobility, and the number of hydrogen bonding patterns. The thermodynamic stability and the conformational characteristics of tetrahomodioxacalix[4]arenes are discussed in regards of the number of phenolic hydrogen bonding patterns and the polarity of a molecule. The substituent effects on the para position of the phenolic ring are also introduced.

• Journal of applied mathematics & informatics
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• v.38 no.3_4
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• pp.271-276
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• 2020

Let R be a commutative ring with identity, R[X] the polynomial ring over R, ∗ a radical operation on R and ⋆ a radical operation of finite character on R[X]. In this paper, we give Hilbert basis theorem for rings with ∗-Noetherian spectrum. More precisely, we show that if (I^∗R[X])^⋆ = (IR[X])^⋆ and (I^∗R[X])^⋆ ∩ R = I^∗ for all ideals I of R, then R has ∗-Noetherian spectrum if and only if R[X] has ⋆-Noetherian spectrum. This is a generalization of a well-known fact that R has Noetherian spectrum if and only if R[X] has Noetherian spectrum.

• Communications of the Korean Mathematical Society
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• v.34 no.4
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• pp.1099-1104
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• 2019

In [1], Ali and Dar proved the ${\ast}$-version of classical theorem due to Posner [15, Theorem] with involution of the second kind. The main objective of this paper is to improve the above mentioned result without the condition of the second kind involution. Moreover, a related result has been discussed.