Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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PROJECTIVE AND INJECTIVE PROPERTIES OF REPRESENTATIONS OF A QUIVER Q = • → • → •

  • Park, Sangwon (Department of Mathematics Dong-A University) ;
  • Han, Juncheol (Department of Mathematics Educations Pusan National University)
  • Received : 2009.07.28
  • Published : 2009.06.30
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Abstract

We define injective and projective representations of a quiver $Q={\bullet}{\rightarrow}{\bullet}{\rightarrow}{\bullet}$. Then we show that a representation $M_1\longrightarrow[50]^{f1}M_2\longrightarrow[50]^{f2}M_3$ of a quiver $Q={\bullet}{\rightarrow}{\bullet}{\rightarrow}{\bullet}$ is projective if and only if each $M_1,\;M_2,\;M_3$ is projective left R-module and $f_1(M_1)$ is a summand of $M_2$ and $f_2(M_2)$ is a summand of $M_3$. And we show that a representation $M_1\longrightarrow[50]^{f1}M_2\longrightarrow[50]^{f2}M_3$ of a quiver $Q={\bullet}{\rightarrow}{\bullet}{\rightarrow}{\bullet}$ is injective if and only if each $M_1,\;M_2,\;M_3$ is injective left R-module and $ker(f_1)$ is a summand of $M_1$ and $ker(f_2)$ is a summand of $M_2$.

Keywords

Acknowledgement

Supported by : Dong-A University

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