• Kyungpook Mathematical Journal
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• v.18 no.1
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• pp.105-107
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• 1978

Considering the prime z-filters on a topological space X through the structures of the ring C(X) of continuous functions. a prime z-filter is uniquely determined by a primary z-ideal in the ring C(X), i. e., they have a one-to-one correspondence. Any primary ideal is contained in a unique maximal ideal in C(X). Denoting $\mathfrak{F}(X)$, $\mathfrak{Q}(X)$, 𝔐(X) the prime, primary-z, maximal spectra, respectively, $\mathfrak{Q}(X)$ is neither an open nor a closed subspace of $\mathfrak{F}(X)$.

• Bulletin of the Korean Mathematical Society
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• v.39 no.3
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• pp.411-422
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• 2002

In this note we are concerned with relationships between one-sided ideals and two-sided ideals, and study the properties of polynomial rings whose maximal one-sided ideals are two-sided, in the viewpoint of the Nullstellensatz on noncommutative rings. Let R be a ring and R[x] be the polynomial ring over R with x the indeterminate. We show that eRe is right quasi-duo for $0{\neq}e^2=e{\in}R$ if R is right quasi-duo; R/J(R) is commutative with J(R) the Jacobson radical of R if R[$\chi$] is right quasi-duo, from which we may characterize polynomial rings whose maximal one-sided ideals are two-sided; if R[x] is right quasi-duo then the Jacobson radical of R[x] is N(R)[x] and so the $K\ddot{o}the's$ conjecture (i.e., the upper nilradical contains every nil left ideal) holds, where N(R) is the set of all nilpotent elements in R. Next we prove that if the polynomial rins R[x], over a reduced ring R with $\mid$X$\mid$ $\geq$ 2, is right quasi-duo, then R is commutative. Several counterexamples are included for the situations that occur naturally in the process of this note.

• 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 Mathematical Society
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• v.22 no.1
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• pp.1-5
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• 1985

The differential polynomial ring A[X, D] has been studied by many authors J. Cozzens, C. Faith, R.E. Johnson and D. Mathis and others. The main purpose of the present paper is to study some properties of .phi.-differential polynomial ring A[X, D, .phi.] and modules over the .phi.-differential polynomial ring X[X, D, .phi.].

• Journal of the Korean Mathematical Society
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• v.47 no.3
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• pp.455-466
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• 2010

For an endomorphism $\alpha$ of a ring R, we introduce the weak $\alpha$-skew Armendariz rings which are a generalization of the $\alpha$-skew Armendariz rings and the weak Armendariz rings, and investigate their properties. Moreover, we prove that a ring R is weak $\alpha$-skew Armendariz if and only if for any n, the $n\;{\times}\;n$ upper triangular matrix ring $T_n(R)$ is weak $\bar{\alpha}$-skew Armendariz, where $\bar{\alpha}\;:\;T_n(R)\;{\rightarrow}\;T_n(R)$ is an extension of $\alpha$ If R is reversible and $\alpha$ satisfies the condition that ab = 0 implies $a{\alpha}(b)=0$ for any a, b $\in$ R, then the ring R[x]/($x^n$) is weak $\bar{\alpha}$-skew Armendariz, where ($x^n$) is an ideal generated by $x^n$, n is a positive integer and $\bar{\alpha}\;:\;R[x]/(x^n)\;{\rightarrow}\;R[x]/(x^n)$ is an extension of $\alpha$. If $\alpha$ also satisfies the condition that ${\alpha}^t\;=\;1$ for some positive integer t, the ring R[x] (resp, R[x; $\alpha$) is weak $\bar{\alpha}$-skew (resp, weak) Armendariz, where $\bar{\alpha}\;:\;R[x]\;{\rightarrow}\;R[x]$ is an extension of $\alpha$.

• Kyungpook Mathematical Journal
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• v.53 no.3
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• pp.397-406
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• 2013

A ring R is called a left (right) SF-ring if simple left (right) R-modules are flat. It is still unknown whether a left (right) SF-ring is von Neumann regular. In this paper, we give some conditions for a left (right) SF-ring to be (a) von Neumann regular; (b) strongly regular; (c) division ring. It is proved that: (1) a right SF-ring R is regular if maximal essential right (left) ideals of R are weakly left (right) ideals of R (this result gives an affirmative answer to the question raised by Zhang in 1994); (2) a left SF-ring R is strongly regular if every non-zero left (right) ideal of R contains a non-zero left (right) ideal of R which is a W-ideal; (3) if R is a left SF-ring such that $l(x)(r(x))$ is an essential left (right) ideal for every right (left) zero divisor x of R, then R is a division ring.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.32 no.1
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• pp.35-43
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• 1995

IF R is a left Artinian ring with identity, G is the group of units of R and X is the set of nonzero, nonunits of R, then G acts naturally on X by conjugation. It is shown that if the conjugate action on X by G is trivial, that is, gx = xg for all $g \in G$ and all $x \in X$, then R is a commutative ring. It is also shown that if the conjegate action on X by G is transitive, then R is a local ring and $J^2 = (0)$ where J is the Jacobson radical of R. In addition, if G is a simple group, then R is isomorphic to $Z_2 [x]/(x^2 + 1) or Z_4$.

• Journal of applied mathematics & informatics
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• v.40 no.3_4
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• pp.709-717
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• 2022

The objective of this research is to prove that an additive mapping T : R → R is a left as well as right centralizer on R if it satisfies any one of the following identities: (i) T(xⁿyⁿ + yⁿxⁿ) = T(xⁿ)yⁿ + yⁿT(xⁿ) (ii) 2T(xⁿyⁿ) = T(xⁿ)yⁿ + yⁿT(xⁿ) for each x, y ∈ R, where n ≥ 1 is a fixed integer and R is any n!-torsion free semiprime ring. In addition, we talk over above identities in the setting of *-ring(ring with involution).

• Korean Journal of Mathematics
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• v.20 no.3
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• pp.273-284
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• 2012

A ring R is called reversibly Armendariz if $b_ja_i=0$ for all $i$, $j$ whenever $f(x)g(x)=0$ for two polynomials $f(x)=\sum_{i=0}^{m}a_ix^i,\;g(x)=\sum_{j=0}^{n}b_jx^j$ over R. It is proved that a ring R is reversibly Armendariz if and only if its polynomial ring is reversibly Armendariz if and only if its Laurent polynomial ring is reversibly Armendariz. Relations between reversibly Armendariz rings and related ring properties are examined in this note, observing the structures of many examples concerned. Various kinds of reversibly Armendariz rings are provided in the process. Especially it is shown to be possible to construct reversibly Armendariz rings from given any Armendariz rings.