• Title, Summary, Keyword: Ring of Polynomials

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IRREDUCIBILITY OF HURWITZ POLYNOMIALS OVER THE RING OF INTEGERS

  • Oh, Dong Yeol;Seo, Ye Lim
    • Korean Journal of Mathematics
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    • v.27 no.2
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    • pp.465-474
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    • 2019
  • Let ${\mathbb{Z}}$ be the ring of integers and ${\mathbb{Z}}[X]$ (resp., $h({\mathbb{Z}})$) be the ring of polynomials (resp., Hurwitz polynomials) over ${\mathbb{Z}}$. In this paper, we study the irreducibility of Hurwitz polynomials in $h({\mathbb{Z}})$. We give a sufficient condition for Hurwitz polynomials in $h({\mathbb{Z}})$ to be irreducible, and we then show that $h({\mathbb{Z}})$ is not isomorphic to ${\mathbb{Z}}[X]$. By using a relation between usual polynomials in ${\mathbb{Z}}[X]$ and Hurwitz polynomials in $h({\mathbb{Z}})$, we give a necessary and sufficient condition for Hurwitz polynomials over ${\mathbb{Z}}$ to be irreducible under additional conditions on the coefficients of Hurwitz polynomials.

ANNIHILATORS IN ONE-SIDED IDEALS GENERATED BY COEFFICIENTS OF ZERO-DIVIDING POLYNOMIALS

  • Kwak, Tai Keun;Lee, Dong Su;Lee, Yang
    • Journal of the Korean Mathematical Society
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    • v.51 no.3
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    • pp.495-507
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    • 2014
  • Nielsen and Rege-Chhawchharia called a ring R right McCoy if given nonzero polynomials f(x), g(x) over R with f(x)g(x) = 0, there exists a nonzero element r ${\in}$ R with f(x)r = 0. Hong et al. called a ring R strongly right McCoy if given nonzero polynomials f(x), g(x) over R with f(x)g(x) = 0, f(x)r = 0 for some nonzero r in the right ideal of R generated by the coefficients of g(x). Subsequently, Kim et al. observed similar conditions on linear polynomials by finding nonzero r's in various kinds of one-sided ideals generated by coefficients. But almost all results obtained by Kim et al. are concerned with the case of products of linear polynomials. In this paper we examine the nonzero annihilators in the products of general polynomials.

AN ELABORATION OF ANNIHILATORS OF POLYNOMIALS

  • Cheon, Jeoung Soo;Kim, Hong Kee;Kim, Nam Kyun;Lee, Chang Ik;Lee, Yang;Sung, Hyo Jin
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.2
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    • pp.521-541
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    • 2017
  • In this note we elaborate first on well-known theorems for annihilators of polynomials over IFP rings by investigating the concrete shapes of nonzero constant annihilators. We consider next a generalization of IFP which preserves Abelian property, in relation with annihilators of polynomials, observing the basic structure of rings satisfying such condition.

ON SOME TYPE ELEMENTS OF ZERO-SYMMETRIC NEAR-RING OF POLYNOMIALS

  • Hashemi, Ebrahim;Shokuhifar, Fatemeh
    • Journal of the Korean Mathematical Society
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    • v.56 no.1
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    • pp.183-195
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    • 2019
  • Let R be a commutative ring with unity. In this paper, we characterize the unit elements, the regular elements, the ${\pi}$-regular elements and the clean elements of zero-symmetric near-ring of polynomials $R_0[x]$, when $nil(R)^2=0$. Moreover, it is shown that the set of ${\pi}$-regular elements of $R_0[x]$ forms a semigroup. These results are somewhat surprising since, in contrast to the polynomial ring case, the near-ring of polynomials has substitution for its "multiplication" operation.

RIGIDNESS AND EXTENDED ARMENDARIZ PROPERTY

  • Baser, Muhittin;Kaynarca, Fatma;Kwak, Tai-Keun
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.1
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    • pp.157-167
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    • 2011
  • For a ring endomorphism of a ring R, Krempa called $\alpha$ rigid endomorphism if $a{\alpha}(a)$ = 0 implies a = 0 for a $\in$ R, and Hong et al. called R an $\alpha$-rigid ring if there exists a rigid endomorphism $\alpha$. Due to Rege and Chhawchharia, a ring R is called Armendariz if whenever the product of any two polynomials in R[x] over R is zero, then so is the product of any pair of coefficients from the two polynomials. The Armendariz property of polynomials was extended to one of skew polynomials (i.e., $\alpha$-Armendariz rings and $\alpha$-skew Armendariz rings) by Hong et al. In this paper, we study the relationship between $\alpha$-rigid rings and extended Armendariz rings, and so we get various conditions on the rings which are equivalent to the condition of being an $\alpha$-rigid ring. Several known results relating to extended Armendariz rings can be obtained as corollaries of our results.

ON ANNIHILATOR IDEALS OF A NEARRING OF SKEW POLYNOMIALS OVER A RING

  • Hashemi, Ebrahim
    • Journal of the Korean Mathematical Society
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    • v.44 no.6
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    • pp.1267-1279
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    • 2007
  • For a ring endomorphism ${\alpha}$ and an ${\alpha}-derivation\;{\delta}$ of a ring R, we study relation between the set of annihilators in R and the set of annihilators in nearring $R[x;{\alpha},{\delta}]\;and\;R_0[[x;{\alpha}]]$. Also we extend results of Armendariz on the Baer and p.p. conditions in a polynomial ring to certain analogous annihilator conditions in a nearring of skew polynomials. These results are somewhat surprising since, in contrast to the skew polynomial ring and skew power series case, the nearring of skew polynomials and skew power series have substitution for its "multiplication" operation.

A STRUCTURE ON COEFFICIENTS OF NILPOTENT POLYNOMIALS

  • Jeon, Young-Cheol;Lee, Yang;Ryu, Sung-Ju
    • Journal of the Korean Mathematical Society
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    • v.47 no.4
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    • pp.719-733
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    • 2010
  • We observe a structure on the products of coefficients of nilpotent polynomials, introducing the concept of n-semi-Armendariz that is a generalization of Armendariz rings. We first obtain a classification of reduced rings, proving that a ring R is reduced if and only if the n by n upper triangular matrix ring over R is n-semi-Armendariz. It is shown that n-semi-Armendariz rings need not be (n+1)-semi-Armendariz and vice versa. We prove that a ring R is n-semi-Armendariz if and only if so is the polynomial ring over R. We next study interesting properties and useful examples of n-semi-Armendariz rings, constructing various kinds of counterexamples in the process.

ON COMMUTATIVITY OF SKEW POLYNOMIALS AT ZERO

  • Jin, Hai-Lan;Kaynarca, Fatma;Kwak, Tai Keun;Lee, Yang
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.1
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    • pp.51-69
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    • 2017
  • We, in this paper, study the commutativity of skew polynomials at zero as a generalization of an ${\alpha}-rigid$ ring, introducing the concept of strongly skew reversibility. A ring R is be said to be strongly ${\alpha}-skew$ reversible if the skew polynomial ring $R[x;{\alpha}]$ is reversible. We examine some characterizations and extensions of strongly ${\alpha}-skew$ reversible rings in relation with several ring theoretic properties which have roles in ring theory.

A NEW CLASS OF CYCLIC CODES USING ORDERED POWER PRODUCT OF POLYNOMIALS

  • Gaur, Ankita;Sharma, Bhudev
    • Journal of applied mathematics & informatics
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    • v.32 no.3_4
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    • pp.529-537
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    • 2014
  • The paper introduces a new product of polynomials defined over a field. It is a generalization of the ordinary product with inner polynomial getting non-overlapping segments obtained by multiplying with coefficients and variable with expanding powers. It has been called 'Ordered Power Product' (OPP). Considering two rings of polynomials $R_m[x]=F[x]modulox^m-1$ and $R_n[x]=F[x]modulox^n-1$, over a field F, the paper then considers the newly introduced product of the two polynomial rings. Properties and algebraic structure of the product of two rings of polynomials are studied and it is shown to be a ring. Using the new type of product of polynomials, we define a new product of two cyclic codes and devise a method of getting a cyclic code from the 'ordered power product' of two cyclic codes. Conditions for the OPP of the generators polynomials of component codes, giving a cyclic code are examined. It is shown that OPP cyclic code so obtained is more efficient than the one that can be obtained by Kronecker type of product of the same component codes.

MCCOY CONDITION ON IDEALS OF COEFFICIENTS

  • Cheon, Jeoung Soo;Huh, Chan;Kwak, Tai Keun;Lee, Yang
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.6
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    • pp.1887-1903
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    • 2013
  • We continue the study of McCoy condition to analyze zero-dividing polynomials for the constant annihilators in the ideals generated by the coefficients. In the process we introduce the concept of ideal-${\pi}$-McCoy rings, extending known results related to McCoy condition. It is shown that the class of ideal-${\pi}$-McCoy rings contains both strongly McCoy rings whose non-regular polynomials are nilpotent and 2-primal rings. We also investigate relations between the ideal-${\pi}$-McCoy property and other standard ring theoretic properties. Moreover we extend the class of ideal-${\pi}$-McCoy rings by examining various sorts of ordinary ring extensions.