IJPAM: Volume 60, No. 1 (2010)

WEAKLY GOLDIE EXTENDING MODULES

Fangdi Kong
Department of Applied Mathematics
Lanzhou University of Technology
Lanzhou, Gansu, 730050, P.R. CHINA
e-mail: [email protected]


Abstract.As a proper generalization of Goldie extending module, we introduce the concept of weakly $\mathscr{G}$-extending module. Let $X,Y\leq M$. Then $X\gamma Y$ if and only if there exist submodules $A$ and $A'$ of $M$ such that $A\cong A'$, $A\leq_eX$ and $A'\leq_eY$. $M$ is called a weakly $\mathscr{G}$-extending module if for every submodule $X$ of $M$, there exists a direct summand $D$ of $M$ such that $X\gamma D$. Let $M=M_1\oplus M_2$, where $M_1$ and $M_2$ are weakly $\mathscr{G}$-extending. It is shown that if $M_1$ is $M_2$-ejective (or $M_2$ is $M_1$-ejective), then $M$ is weakly $\mathscr{G}$-extending. Finally, we show that if $M$ is weakly $\mathscr{G}$-extending, then so is its rational hull.

Received: September 19, 2009

AMS Subject Classification: 16D70

Key Words and Phrases: extending module, $\mathscr{G}$-extending module, weakly $\mathscr{G}$-extending module

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 60
Issue: 1