Document Type : Research Paper

Authors

• Amenah Hasan
• Wasan Khalid

Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq

Abstract

For any $R$-module $W,\ D_j(W)$ presented as the total of all $J$-small sub-modules. If $A$ and $B$ are sub-module of $W$, we say $A$ is $\bigoplus \ D_j$'supplement of $B$ in $W$ if $W=A+B=A\bigoplus \acute{A}$, for $\acute{A}\underline{\hookrightarrow}W$, and $A\bigcap B\ll_j D_j(A)$. If every sub-module has $\bigoplus \ D_j$-supplemented, then $W$ is $\bigoplus \ D_j$-supplemented $A$ sub-module $A$ of $W$. If a sentence is conclusive, it is said to be cofinite i.e., $\frac{W}{A}$ is finitely generated. Also we introduce cofinite $\bigoplus \ D_j$-supplemented if every cofinite sub-module of $W$ has $\bigoplus \ D_j$-supplemented.

Keywords

• Cofinite
• finitely generated
• module
• supplemented module