• Journal of the Korean Mathematical Society
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• v.47 no.5
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• pp.879-897
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• 2010

Y. Hirano introduced the concept of a quasi-Armendariz ring which extends both Armendariz rings and semiprime rings. A ring R is called quasi-Armendariz if $a_iRb_j$ = 0 for each i, j whenever polynomials $f(x)\;=\;\sum_{i=0}^ma_ix^i$, $g(x)\;=\;\sum_{j=0}^mb_jx^j\;{\in}\;R[x]$ satisfy f(x)R[x]g(x) = 0. In this paper, we first extend the quasi-Armendariz property of semiprime rings to the skew polynomial rings, that is, we show that if R is a semiprime ring with an epimorphism $\sigma$, then f(x)R[x; $\sigma$]g(x) = 0 implies $a_iR{\sigma}^{i+k}(b_j)=0$ for any integer k $\geq$ 0 and i, j, where $f(x)\;=\;\sum_{i=0}^ma_ix^i$, $g(x)\;=\;\sum_{j=0}^mb_jx^j\;{\in}\;R[x,\;{\sigma}]$. Moreover, we extend this property to the skew monoid rings, the Ore extensions of several types, and skew power series ring, etc. Next we define $\sigma$-skew quasi-Armendariz rings for an endomorphism $\sigma$ of a ring R. Then we study several extensions of $\sigma$-skew quasi-Armendariz rings which extend known results for quasi-Armendariz rings and $\sigma$-skew Armendariz rings.

• Bulletin of the Korean Mathematical Society
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• v.41 no.3
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• pp.507-519
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• 2004

For a ring endomorphism $\sigma$ of a ring R, J. Krempa called $\sigma$ a rigid endomorphism if a$\sigma$(a)＝0 implies a＝0 for a ${\in}$R. A ring R is called rigid if there exists a rigid endomorphism of R. In this paper, we extend the (J'-rigid property of a ring R to the upper nilradical $N_{r}$(R) of R. For an endomorphism R and the upper nilradical $N_{r}$(R) of a ring R, we introduce the condition (*): $N_{r}$(R) is a $\sigma$-ideal of R and a$\sigma$(a) ${\in}$ $N_{r}$(R) implies a ${\in}$ $N_{r}$(R) for a ${\in}$ R. We study characterizations of a ring R with an endomorphism $\sigma$ satisfying the condition (*), and we investigate their related properties. The connections between the upper nilradical of R and the upper nilradical of the skew power series ring R[[$\chi$;$\sigma$]] of R are also investigated.ated.

• Kyungpook Mathematical Journal
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• v.63 no.4
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• pp.561-569
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• 2023

An element a ∈ R is strongly rad-clean provided that there exists an idempotent e ∈ R such that a - e ∈ U(R), ae = ea and eae ∈ J(eRe). In this article, we completely determine when a 2 × 2 matrix over a commutative local ring is strongly rad clean. An application to matrices over power-series is also given.

• Journal of the Korean Mathematical Society
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• v.53 no.2
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• pp.447-459
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• 2016

Let D be an integral domain, {$X_{\alpha}$} be a nonempty set of indeterminates over D, and $D{\mathbb{[}}\{X_{\alpha}\}{\mathbb{]}_1}$ be the first type power series ring over D. We show that if D is a t-SFT $Pr{\ddot{u}}fer$ v-multiplication domain, then $D{\mathbb{[}}\{X_{\alpha}\}{\mathbb{]}}_{1_{D-\{0\}}}$ is a Krull domain, and $D{\mathbb{[}}\{X_{\alpha}\}{\mathbb{]}}_1$ is a $Pr{\ddot{u}}fer$ v-multiplication domain if and only if D is a Krull domain.

• Kyungpook Mathematical Journal
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• v.47 no.1
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• pp.127-132
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• 2007

In this paper, we show that if R is a commutative ring with identity and (S, ${\leq}$) is a strictly totally ordered monoid, then the ring [[$R^{S,{\leq}}$]] of generalized power series is a PF-ring if and only if for any two S-indexed subsets A and B of R such that $B{\subseteq}ann_R(|A)$, there exists $c{\in}ann_R(A)$ such that $bc=b$ for all $b{\in}B$, and that for a Noetherian ring R, $[[R^{S,{\leq}}$]] is a PP ring if and only if R is a PP ring.

• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.387-396
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• 2016

An $n{\times}n$ matrix A over a commutative ring is strongly clean provided that it can be written as the sum of an idempotent matrix and an invertible matrix that commute. Let R be an arbitrary commutative ring, and let $A(x){\in}M_n(R[[x]])$. We prove, in this note, that $A(x){\in}M_n(R[[x]])$ is strongly clean if and only if $A(0){\in}M_n(R)$ is strongly clean. Strongly clean matrices over quotient rings of power series are also determined.

• Journal of the Korean Mathematical Society
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• v.49 no.4
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• pp.687-701
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• 2012

Let $R$ be a ring and $nil(R)$ the set of all nilpotent elements of $R$. For a subset $X$ of a ring $R$, we define $N_R(X)=\{a{\in}R{\mid}xa{\in}nil(R)$ for all $x{\in}X$}, which is called a weak annihilator of $X$ in $R$. $A$ ring $R$ is called weak zip provided that for any subset $X$ of $R$, if $N_R(Y){\subseteq}nil(R)$, then there exists a finite subset $Y{\subseteq}X$ such that $N_R(Y){\subseteq}nil(R)$, and a ring $R$ is called weak symmetric if $abc{\in}nil(R){\Rightarrow}acb{\in}nil(R)$ for all a, b, $c{\in}R$. It is shown that a generalized power series ring $[[R^{S,{\leq}}]]$ is weak zip (resp. weak symmetric) if and only if $R$ is weak zip (resp. weak symmetric) under some additional conditions. Also we describe all weak associated primes of the generalized power series ring $[[R^{S,{\leq}}]]$ in terms of all weak associated primes of $R$ in a very straightforward way.

• Kyungpook Mathematical Journal
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• v.62 no.3
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• pp.437-453
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• 2022

Unlike for polynomial rings, the notion of multiplication for the near-ring of polynomials is the substitution operation. This leads to somewhat surprising results. Let S be an abelian left near-ring with identity. The relation ~ on S defined by letting a ~ b if and only if ann_S(a) = ann_S(b), is an equivalence relation. The compressed zero-divisor graph 𝚪_E(S) of S is the undirected graph whose vertices are the equivalence classes induced by ~ on S other than [0]_S and [1]_S, in which two distinct vertices [a]_S and [b]_S are adjacent if and only if ab = 0 or ba = 0. In this paper, we are interested in studying the compressed zero-divisor graphs of the zero-symmetric near-ring of polynomials R₀[x] and the near-ring of the power series R₀[[x]] over a commutative ring R. Also, we give a complete characterization of the diameter of these two graphs. It is natural to try to find the relationship between diam(𝚪_E(R₀[x])) and diam(𝚪_E(R₀[[x]])). As a corollary, it is shown that for a reduced ring R, diam(𝚪_E(R)) ≤ diam(𝚪_E(R₀[x])) ≤ diam(𝚪_E(R₀[[x]])).

• Journal of the Chungcheong Mathematical Society
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• v.32 no.4
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• pp.401-408
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• 2019

A ring R is called right (resp., left) nilpotent-duo if N(R)a ⊆ aN(R) (resp., aN(R) ⊆ N(R)a) for every a ∈ R, where N(R) is the set of all nilpotents in R. In this article we provide other proofs of known results and other computations for known examples in relation with right nilpotent-duo property. Furthermore we show that the left nilpotent-duo property does not go up to a kind of matrix ring.

• Journal of the Korean Mathematical Society
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• v.55 no.1
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• pp.101-129
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• 2018

From an analytical perspective, we introduce a sequence of Cartier operators that act on the field of formal Laurent series in one variable with coefficients in a field of positive characteristic p. In this work, we discover the binomial inversion formula between Hasse derivatives and Cartier operators, implying that Cartier operators can play a prominent role in various objects of study in function field arithmetic, as a suitable substitute for higher derivatives. For an applicable object, the Wronskian criteria associated with Cartier operators are introduced. These results stem from a careful study of two types of Cartier operators on the power series ring ${\mathbf{F}}_q$[[T]] in one variable T over a finite field ${\mathbf{F}}_q$ of q elements. Accordingly, we show that two sequences of Cartier operators are an orthonormal basis of the space of continuous ${\mathbf{F}}_q$-linear functions on ${\mathbf{F}}_q$[[T]]. According to the digit principle, every continuous function on ${\mathbf{F}}_q$[[T]] is uniquely written in terms of a q-adic extension of Cartier operators, with a closed-form of expansion coefficients for each of the two cases. Moreover, the p-adic analogues of Cartier operators are discussed as orthonormal bases for the space of continuous functions on ${\mathbf{Z}}_p$.