• Title/Summary/Keyword: nil algebra

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NILPOTENCY INDEX OF NIL-ALGEBRA OF NIL-INDEX 3

  • LEE WOO
    • Journal of applied mathematics & informatics
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    • v.20 no.1_2
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    • pp.569-573
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    • 2006
  • Nagata and Higman proved that any nil-algebra of finite nilindex is nilpotent of finite index. The Nagata-Higman Theorem can be formulated in terms of T-ideals. TheT-ideal generated by $a^n$ for all $a{\in}A$ is also generated by the symmetric polynomials. The symmetric polynomials play an importmant role in analyzing nil-algebra. We construct the incidence matrix with the symmetric polynomials. Using this incidence matrix, we determine the nilpotency index of nil-algebra of nil-index 3.

DETERMINANT OF INCIDENCE MATRIX OF NIL-ALGEBRA

  • Lee, Woo
    • Communications of the Korean Mathematical Society
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    • v.17 no.4
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    • pp.577-581
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    • 2002
  • The incidence matrices corresponding to a nil-algebra of finite index % can be used to determine the nilpotency. We find the smallest positive integer n such that the sum of the incidence matrices Σ$\_$p/$\^$p/ is invertible. In this paper, we give a different proof of the case that the nil-algebra of index 2 has nilpotency less than or equal to 4.

NIL SUBSETS IN BCH-ALGEBRAS

  • Jun, Young-Bae;Roh, Eun-Hwan
    • East Asian mathematical journal
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    • v.22 no.2
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    • pp.207-213
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    • 2006
  • Using the notion of nilpotent elements, the concept of nil subsets is introduced, and related properties are investigated. We show that a nil subset on a subalgebra (resp. (closed) ideal) is a subalgebra (resp. (closed) ideal). We also prove that in a nil algebra every ideal is a subalgebra.

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k-NIL RADICAL IN BCI-ALGEBRAS II

  • Jun, Y.B;Hong, S.M
    • Communications of the Korean Mathematical Society
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    • v.12 no.3
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    • pp.499-505
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    • 1997
  • This paper is a continuation of [3]. We prove that if A is quasi-associative (resp. an implicative) ideal of a BCI-algebra X then the k-nil radical of A is a quasi-associative (resp. an implicative) ideal of X. We also construct the quotient algebra $X/[Z;k]$ of a BCI-algebra X by the k-nhil radical [A;k], and show that if A and B are closed ideals of BCI-algebras X and Y respectively, then

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THE RANGE INCLUSION RESULTS FOR ALGEBRAIC NIL DERIVATIONS ON COMMUTATIVE AND NONCOMMUTATIVE ALGEBRAS

  • Toumi, Mohamed Ali
    • The Pure and Applied Mathematics
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    • v.20 no.4
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    • pp.243-249
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    • 2013
  • Let A be an algebra and D a derivation of A. Then D is called algebraic nil if for any $x{\in}A$ there is a positive integer n = n(x) such that $D^{n(x)}(P(x))=0$, for all $P{\in}\mathbb{C}[X]$ (by convention $D^{n(x)}({\alpha})=0$, for all ${\alpha}{\in}\mathbb{C}$). In this paper, we show that any algebraic nil derivation (possibly unbounded) on a commutative complex algebra A maps into N(A), where N(A) denotes the set of all nilpotent elements of A. As an application, we deduce that any nilpotent derivation on a commutative complex algebra A maps into N(A), Finally, we deduce two noncommutative versions of algebraic nil derivations inclusion range.

RADICALS AND HOMOMORPHIC IMAGES OF ${C^*}$-ALGEBRAS

  • Han, Hyuk
    • Communications of the Korean Mathematical Society
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    • v.14 no.2
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    • pp.365-371
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    • 1999
  • In this paper, we prove that the range of homomorphism from a C\ulcorner-algebra A into a commutative Banach algebra B whose radical is nil contains no non-zero element of the radical of B. Using this result we show that there is no non-zero homomorphism from a C\ulcorner-algebra into a commutative radical nil Banach algebra.

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ON THE RANGE OF DERIVATIONS

  • Chang, Ick-Soon
    • Journal of the Chungcheong Mathematical Society
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    • v.12 no.1
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    • pp.187-191
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    • 1999
  • In this paper we will show that if [G(y), x]D(x) lies in the nil radical of A for all $x{\in}A$, then GD maps A into the radical, where D and G are derivations on a Banach algebra A.

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RESULTS ON THE RANGE OF DERIVATIONS

  • Jung, Yong-Soo
    • Bulletin of the Korean Mathematical Society
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    • v.37 no.2
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    • pp.265-272
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    • 2000
  • Let D be a derivation on an Banach algebra A. Suppose that [[D(x), x], D(x)] lies in the nil radical of A for all $x{\;}{\in}{\;}A$. Then D(A) is contained in the Jacobson radical of A.

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A note on k-nil radicals in BCI-algebras

  • Hong, Sung-Min;Xiaolong Xin
    • Bulletin of the Korean Mathematical Society
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    • v.34 no.2
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    • pp.205-209
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    • 1997
  • Hong et al. [2] and Jun et al. [4] introduced the notion of k-nil radical in a BCI-algebra, and investigated its some properties. In this paper, we discuss the further properties on the k-nil radical. Let A be a subset of a BCI-algebra X. We show that the k-nil radical of A is the union of branches. We prove that if A is an ideal then the k-nil radical [A;k] is a p-ideal of X, and that if A is a subalgebra, then the k-nil radical [A;k] is a closed p-ideal, and hence a strong ideal of X.

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ON NAGATA-HIGMAN THEOREM

  • Lee, Woo
    • Journal of applied mathematics & informatics
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    • v.27 no.5_6
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    • pp.1489-1492
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    • 2009
  • Nagata[3] and Higman[1] showed that nil-algebra of the nilindex n is nilpotent of finite index. In this paper we show that the bounded degree of the nilpotency is less than or equal to $2^n-1$. Our proof needs only some elementary fact about Vandermonde determinant, which is much simpler than Nagata's or Higman's proof.

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