• Title/Summary/Keyword: Irreducible Modules

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A REMARK ON MULTIPLICATION MODULES

  • Choi, Chang-Woo;Kim, Eun-Sup
    • Bulletin of the Korean Mathematical Society
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    • v.31 no.2
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    • pp.163-165
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    • 1994
  • Modules which satisfy the converse of Schur's lemma have been studied by many authors. In [6], R. Ware proved that a projective module P over a semiprime ring R is irreducible if and only if En $d_{R}$(P) is a division ring. Also, Y. Hirano and J.K. Park proved that a torsionless module M over a semiprime ring R is irreducible if and only if En $d_{R}$(M) is a division ring. In case R is a commutative ring, we obtain the following: An R-module M is irreducible if and only if En $d_{R}$(M) is a division ring and M is a multiplication R-module. Throughout this paper, R is commutative ring with identity and all modules are unital left R-modules. Let R be a commutative ring with identity and let M be an R-module. Then M is called a multiplication module if for each submodule N of M, there exists and ideal I of R such that N=IM. Cyclic R-modules are multiplication modules. In particular, irreducible R-modules are multiplication modules.dules.

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STRONGLY IRREDUCIBLE SUBMODULES

  • ATANI, SHAHABADDIN EBRAHIMI
    • Bulletin of the Korean Mathematical Society
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    • v.42 no.1
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    • pp.121-131
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    • 2005
  • This paper is motivated by the results in [6]. We study some properties of strongly irreducible submodules of a module. In fact, our objective is to investigate strongly irreducible modules and to examine in particular when sub modules of a module are strongly irreducible. For example, we show that prime submodules of a multiplication module are strongly irreducible, and a characterization is given of a multiplication module over a Noetherian ring which contain a non-prime strongly irreducible submodule.

Almost Projective Modules over Artin Algebras

  • Park, Jun Seok
    • Journal of the Chungcheong Mathematical Society
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    • v.1 no.1
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    • pp.43-53
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    • 1988
  • The main result of this paper is a characterization of almost projective modules over art in algebras by means of irreducible maps and almost split sequences. A module X is an almost projective module if and only if it has a presentation $0{\longrightarrow}L{\longrightarrow^{\alpha}}P{\longrightarrow}X{\longrightarrow}0$ with projective module P and irreducible maps ${\alpha}$. Let X be an injective almost projective non simple module and $0{\rightarrow}Dtr(x){\rightarrow}E{\rightarrow}X{\rightarrow}0$ be an almost split sequence. If $E=E_1{\oplus}E_2$ is a direct decomposition of indecomposable modules then ${\ell}(X)=3$.

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The Stable Embeddability on Modules over Complex Simple Lie Algebras

  • Kim, Dong-Seok
    • Journal of the Korean Data and Information Science Society
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    • v.18 no.3
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    • pp.827-832
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    • 2007
  • Several partial orders on integral partitions have been studied with many applications such as majorizations, capacities of quantum memory and embeddabilities of matrix algebras. In particular, the embeddability, stable embeddability and strong-stable embeddability problems arise for finite dimensional irreducible modules over a complex simple Lie algebra L. We find a sufficient condition for an L-module strong-stably embeds into another L-module using formal character theory.

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The Embeddability of s1(n,C) Modules

  • Kim, Dong-Seok
    • Journal of the Korean Data and Information Science Society
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    • v.18 no.3
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    • pp.803-808
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    • 2007
  • In present article, we consider the embeddability problems for finite dimensional irreducible modules over a complex simple Lie algebra L. For s1(n,C) modules, we determine when one can be embedded into the other if s1(n,C) modules are tensor products of fundamental modules.

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MODULES WITH PRIME ENDOMORPHISM RINGS

  • Bae, Soon-Sook
    • Journal of the Korean Mathematical Society
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    • v.38 no.5
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    • pp.987-1030
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    • 2001
  • Some discrimination of modules whose endomorhism rings are prime is introduced, by means of structures of submodules inducing prime ideals of the endomorphism ring End(sub)R (M) of a left R-module (sub)RM over a ring R. Modules with non-prime endomorphism rings are contrapositively studied as well.

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SOME PROPERTIES OF VERMA MODULES OVER AFFINE LIE ALGEBRAS

  • Kim, Wan-Soon
    • Communications of the Korean Mathematical Society
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    • v.10 no.4
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    • pp.789-795
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    • 1995
  • For nonintegrable weight $-\rho$, some weight multiplicities of the irreducible module $L(-\rho)$ over $A^{(1)}_{(1)}$ affine Lie algebras are expressed in terms of the colored partition functions. Also we find the multiplicity of $L(-\rho)$ in ther Verma module $M(-\rho)$ for any affine Lie algebras.

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GROBNER-SHIRSHOV BASES FOR IRREDUCIBLE sp4-MODULES

  • Lee, Dong-Il
    • Journal of the Korean Mathematical Society
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    • v.45 no.3
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    • pp.711-725
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    • 2008
  • We give an explicit construction of Grobner-Shirshov pairs and monomial bases for finite-dimensional irreducible representations of the simple Lie algebra $sp_4$. We also identify the monomial basis consisting of the reduced monomials with a set of semistandard tableaux of a given shape, on which we give a colored oriented graph structure.