• Title, Summary, Keyword: polynomial ring

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SOME REMARKS ON SKEW POLYNOMIAL RINGS OVER REDUCED RINGS

  • Kim, Hong-Kee
    • East Asian mathematical journal
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    • v.17 no.2
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    • pp.275-286
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    • 2001
  • In this paper, a skew polynomial ring $R[x;\alpha]$ of a ring R with a monomorphism $\alpha$ are investigated as follows: For a reduced ring R, assume that $\alpha(P){\subseteq}P$ for any minimal prime ideal P in R. Then (i) $R[x;\alpha]$ is a reduced ring, (ii) a ring R is Baer(resp. quasi-Baer, p.q.-Baer, a p.p.-ring) if and only if the skew polynomial ring $R[x;\alpha]$ is Baer(resp. quasi-Baer, p.q.-Baer, a p.p.-ring).

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PROPERTIES OF HURWITZ POLYNOMIAL AND HURWITZ SERIES RINGS

  • Elliott, Jesse;Kim, Hwankoo
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.3
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    • pp.837-849
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    • 2018
  • In this paper, we study the closedness such as seminomality and t-closedness, and Noetherian-like properties such as piecewise Noetherianness and Noetherian spectrum, of Hurwitz polynomial rings and Hurwitz series rings. To do so, we construct an isomorphism between a Hurwitz polynomial ring (resp., a Hurwitz series ring) and a factor ring of a polynomial ring (resp., a power series ring) in a countably infinite number of indeterminates.

SOME RESULTS ON A DIFFERENTIAL POLYNOMIAL RING OVER A REDUCED RING

  • Han, Jun-Cheol;Kim, Hong-Kee;Lee, Yang
    • East Asian mathematical journal
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    • v.16 no.1
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    • pp.89-96
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    • 2000
  • In this paper, a differential polynomial ring $R[x;\delta]$ of ring R with a derivation $\delta$ are investigated as follows: For a reduced ring R, a ring R is Baer(resp. quasi-Baer, p.q.-Baer, p.p.-ring) if and only if the differential polynomial ring $R[x;\delta]$ is Baer(resp. quasi-Baer, p.q.-Baer, p.p.-ring).

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QUASI-ARMENDARIZ PROPERTY FOR SKEW POLYNOMIAL RINGS

  • Baser, Muhittin;Kwa, Tai Keun
    • Communications of the Korean Mathematical Society
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    • v.26 no.4
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    • pp.557-573
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    • 2011
  • The concept of the quasi-Armendariz property of rings properly contains Armendariz rings and semiprime rings. In this paper, we extend the quasi-Armendariz property for a polynomial ring to the skew polynomial ring, hence we call such ring a ${\sigma}$-quasi-Armendariz ring for a ring endomorphism ${\sigma}$, and investigate its structures, several extensions and related properties. In particular, we study the semiprimeness and the quasi-Armendariz property between a ring R and the skew polynomial ring R[x;${\sigma}$$] of R, and so these provide us with an opportunity to study quasi-Armendariz rings and semiprime rings in a general setting, and several known results follow as consequences of our results.

ON JACOBSON AND NIL RADICALS RELATED TO POLYNOMIAL RINGS

  • Kwak, Tai Keun;Lee, Yang;Ozcan, A. Cigdem
    • Journal of the Korean Mathematical Society
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    • v.53 no.2
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    • pp.415-431
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    • 2016
  • This note is concerned with examining nilradicals and Jacobson radicals of polynomial rings when related factor rings are Armendariz. Especially we elaborate upon a well-known structural property of Armendariz rings, bringing into focus the Armendariz property of factor rings by Jacobson radicals. We show that J(R[x]) = J(R)[x] if and only if J(R) is nil when a given ring R is Armendariz, where J(A) means the Jacobson radical of a ring A. A ring will be called feckly Armendariz if the factor ring by the Jacobson radical is an Armendariz ring. It is shown that the polynomial ring over an Armendariz ring is feckly Armendariz, in spite of Armendariz rings being not feckly Armendariz in general. It is also shown that the feckly Armendariz property does not go up to polynomial rings.

MODULES OVER THE $\phi$- DIFFERENTIAL POLYNOMIAL RINGS

  • Sohn, Mun-Gu;Rim, Seog-Hoon
    • 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.].

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RINGS OVER WHICH POLYNOMIAL RINGS ARE ARMENDARIZ AND REVERSIBLE

  • Ahn, Jung Ho;Choi, Min Jeong;Choi, Si Ra;Jeong, Won Seok;Kim, Jung Soo;Lee, Jeong Yeol;Lee, Soon Ji;Lee, Young Sun;Noh, Dong Hyun;Noh, Yu Seung;Park, Gyeong Hyeon;Lee, Chang Ik;Lee, Yang
    • 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.

ON NILPOTENT POWER SERIES WITH NILPOTENT COEFFICIENTS

  • Kwak, Tai Keun;Lee, Yang
    • Korean Journal of Mathematics
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    • v.21 no.1
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    • pp.41-53
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    • 2013
  • Antoine studied conditions which are connected to the question of Amitsur of whether or not a polynomial ring over a nil ring is nil, introducing the notion of nil-Armendariz rings. Hizem extended the nil-Armendariz property for polynomial rings onto power-series rings, say nil power-serieswise rings. In this paper, we introduce the notion of power-serieswise CN rings that is a generalization of nil power-serieswise Armendariz rings. Finally, we study the nil-Armendariz property for Ore extensions and skew power series rings.

IFP RINGS AND NEAR-IFP RINGS

  • Ham, Kyung-Yuen;Jeon, Young-Cheol;Kang, Jin-Woo;Kim, Nam-Kyun;Lee, Won-Jae;Lee, Yang;Ryu, Sung-Ju;Yang, Hae-Hun
    • Journal of the Korean Mathematical Society
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    • v.45 no.3
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    • pp.727-740
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    • 2008
  • A ring R is called IFP, due to Bell, if ab=0 implies aRb=0 for $a,b{\in}R$. Huh et al. showed that the IFP condition need not be preserved by polynomial ring extensions. But it is shown that ${\sum}^n_{i=0}$ $E_{ai}E$ is a nonzero nilpotent ideal of E whenever R is an IFP ring and $0{\neq}f{\in}F$ is nilpotent, where E is a polynomial ring over R, F is a polynomial ring over E, and $a_i^{'s}$ are the coefficients of f. we shall use the term near IFP to denote such a ring as having place near at the IFPness. In the present note the structures of IFP rings and near-IFP rings are observed, extending the classes of them. IFP rings are NI (i.e., nilpotent elements form an ideal). It is shown that the near-IFPness and the NIness are distinct each other, and the relations among them and related conditions are examined.

ANNIHILATING CONTENT IN POLYNOMIAL AND POWER SERIES RINGS

  • Abuosba, Emad;Ghanem, Manal
    • Journal of the Korean Mathematical Society
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    • v.56 no.5
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    • pp.1403-1418
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    • 2019
  • Let R be a commutative ring with unity. If f(x) is a zero-divisor polynomial such that $f(x)=c_f f_1(x)$ with $c_f{\in}R$ and $f_1(x)$ is not zero-divisor, then $c_f$ is called an annihilating content for f(x). In this case $Ann(f)=Ann(c_f )$. We defined EM-rings to be rings with every zero-divisor polynomial having annihilating content. We showed that the class of EM-rings includes integral domains, principal ideal rings, and PP-rings, while it is included in Armendariz rings, and rings having a.c. condition. Some properties of EM-rings are studied and the zero-divisor graphs ${\Gamma}(R)$ and ${\Gamma}(R[x])$ are related if R was an EM-ring. Some properties of annihilating contents for polynomials are extended to formal power series rings.