\section{“‰ã‰À ğ‰Ü‰Àı} \vspace{-1cm}
{\defin.\label{d3-1}} õ‰¹‰Ş‰ä‰‚ \InE{}$\{A_i : i\in I\}$\EnE{} ¥ ¥ş‰Âõ‰Àøñû‰‘ı ÷‰‘¬‰Ô‰Â \InE{}$M$\EnE{} ¤ ¡‰‘÷‰¢ùı
õ‰Æ‰µ‰Ö‰Û ¥ ¥ş‰Âõ‰Àøñû‰‘ õ‰ü÷‰‘õ‰ƒ‰İ, û‰Â ğ‰‘ù “‰Âı û‰Â ¥ş‰Â õ‰¹‰Ş‰ä‰‚ õ‰µ‰€‰‘û‰ü \InE{}$J\subseteq I$\EnE{}
ø “‰Âı û‰Â \InE{}$i\in I-J$\EnE{}, \InE{}$(\sum_{j\in J} A_j)\cap {A_i}=(0)$\EnE{}.
{\lem.\label{l3-1}} ê‰Â­ ‰ƒ‰İ  \InE{}$\{A_i : i\in I\}$\EnE{} ¡‰‘÷‰¢ùı õ‰Æ‰µ‰Ö‰Û ¥ ¥ş‰Âõ‰Àøñû‰‘ı \InE{}$M$\EnE{} “‰‘ª‰À ø \InE{}$A\leq M$\EnE{}. ¢¤ ş‰ß ¬‰¤– ğ‰Â “‰Âı û‰Â ¥ş‰Â õ‰¹‰Ş‰ä‰‚ õ‰µ‰€‰‘û‰ü \InE{}$J$\EnE{} ¥ \InE{}$I$\EnE{},
\InE{}$A\cap (\sum_{j\in J} A_j)=(0)$\EnE{}, öğ‰‘ù \InE{}$\{A_i : i\in I\}\cup \{A\}$\EnE{} ÷‰ƒ‰Ã
¡‰‘÷‰¢ùı õ‰Æ‰µ‰Ö‰Û ¥ ¥ş‰Âõ‰Àøñû‰‘¨‰´. 

{\proof} “‰Àş‰ú‰ü ¨‰´. \InE{}$\blk$\EnE{}.
{\tab.\label{s3-1}} “‰‘ ¨‰µ‰Ô‰‘¢ù ¥ ó‰İ —‰Æ‰¤ö õ‰ü—‰ö ğ‰Ô‰´ ğ‰Â  \InE{}$X=\{A_i : i\in I\}$\EnE{}
¡‰‘÷‰¢ùı õ‰Æ‰µ‰Ö‰Û ¥ ¥ş‰Âõ‰Àøñû‰‘ı \InE{}$-R$\EnE{}õ‰Àøñ \InE{}$M$\EnE{} “‰‘ª‰À, öğ‰‘ù õ‰¹‰Ş‰ä‰‚ 
\InE{}$Y=\{B_j : B_j\subseteq M\}$\EnE{} ø›‰¢ ¢¤¢ “‰‚ ω¤ı  “‰Âı \InE{}$j\in J$\EnE{}, \InE{}$B_j=A_j$\EnE{}
 ø \InE{}$Y$\EnE{} ¡‰‘÷‰¢ù õ‰Æ‰µ‰Ö‰Û õ‰‘î‰Æ‰ƒ‰Ş‰‘ñ “‰‘ª‰À. “‰‘ ¨‰µ‰Ô‰‘¢ù ¥ ó‰İ \ref{l3-1}, ğ‰Â \InE{}$A$\EnE{} ¥ş‰Âõ‰Àøñ
 ÷‰‘¬‰Ô‰Âı ¥ \InE{}$M$\EnE{} “‰‘ª‰À ø \InE{}$A\notin Y$\EnE{}, öğ‰‘ù ¥ş‰Âõ‰¹‰Ş‰ä‰‚ õ‰µ‰€‰‘û‰ü \InE{}$K$\EnE{} ¥ \InE{}$J$\EnE{} ø›‰¢ 
 ¢¤¢ “‰‚ ω¤ı  \InE{}$(\sum_{i\in K}A_i)\cap A \neq(0)$\EnE{}.
{\defin.\label{d3-2}} “‰ã‰À ğ‰Ü‰Àı \InE{}$-R$\EnE{}õ‰Àøñ ¤¨‰´  \InE{}$M$\EnE{} ¤  “‰‘ \InE{}$G-dimM_R$\EnE{} ÷‰È‰‘ö õ‰ü¢û‰ƒ‰İ ‰Ø‰µ‰Âş‰ß ä‰À¢
 ¬‰Ü‰ü ¨‰´  “‰Ã¤ğ‰µ‰Â ş‰‘ õ‰Æ‰‘øı ä‰À¢ ¬‰Ü‰ü —‰Ş‰‘ô ¡‰‘÷‰¢ùû‰‘ı õ‰Æ‰µ‰Ö‰Û ¥ş‰Âõ‰Àøñû‰‘ı \InE{}$M$\EnE{}
 ¨‰´. \InE{}$G-dimR$\EnE{} ÷‰Ş‰‘ş‰‘÷‰Ú‰Â “‰ã‰À ğ‰Ü‰Àı Ÿ‰Ü‰Ö‰‚ \InE{}$R$\EnE{} ¨‰´ ø쉵‰ü  “‰‚ 䉀‰ö \InE{}$-R$\EnE{}õ‰Àøñ  ¤¨‰´ 
 ¢¤ ÷‰Ñ‰Â ğ‰Âꉵ‰‚ ª‰¢.
{\defin.\label{d2-8}} Ÿ‰Ü‰Ö‰‚ \InE{}$R$\EnE{} “‰‘ “‰ã‰Àğ‰Ü‰Àı õ‰µ‰€‰‘û‰ü ¤¨‰´ ÷‰‘õ‰ƒ‰Àù õ‰üª‰¢, û‰Â ğ‰‘ù \InE{}$R$\EnE{} ª‰‘õ‰Û û‰ƒ‰º ¡‰‘÷‰¢ù ÷‰‘õ‰µ‰€‰‘û‰ü õ‰Æ‰µ‰Ö‰Û ¥ ş‰Àñû‰‘ı ¤¨‰´ ÷‰‘¬‰Ô‰Â ÷‰±‰‘ª‰À.
{\defin.\label{d2-9}} Ÿ‰Ü‰Ö‰‚ \InE{}$R$\EnE{} Ÿ‰Ü‰Ö‰‚ı ğ‰Ü‰Àı ¤¨‰´ ÷‰‘õ‰ƒ‰Àù õ‰üª‰¢, û‰Â ğ‰‘ù \InE{}$R$\EnE{} ¢¤ı “‰ã‰Àğ‰Ü‰Àı õ‰µ‰€‰‘û‰ü ¤¨‰´ “‰‘ª‰À ø ¢¤ ª‰Â¯ \InE{}$ACC$\EnE{}
 ¤øı •‰‰Ú‰Âû‰‘ı ¤¨‰´ ¬‰Àë ‰À. “‰‚ 䉀‰ö õ‰·‰‘ñ û‰Â Ÿ‰Ü‰Ö‰‚ı ÷‰ş‰µ‰Âı ¤¨‰´, ğ‰Ü‰Àı ¤¨‰´ ¨‰´. 
{\coro.\label{c3-1}}  \InE{}$G-dimM_R=1$\EnE{} ğ‰Â ø —‰€‰ú‰‘ ğ‰Â \InE{}$-R$\EnE{}õ‰Àøñ \InE{}$M$\EnE{} ş‰Ø‰€‰¡‰´ “‰‘ª‰À.

{\proof} “‰Àş‰ú‰ü ¨‰´. \InE{}$\blk$\EnE{}
{\theo)š-ø÷‰Å-\kasre ñ ꉁî‰Å(.\label{t3-1}} Ÿ‰Ü‰Ö‰‚ı “‰‘ “‰ã‰À ğ‰Ü‰Àı ¢ó‰¿‰ù ø›‰¢ ¢¤¢.
 
 {\proof}¤.í. [6].\InE{}$\blk$\EnE{}
 {\theo.\label{t3-2}} õ‰Àøñ \InE{}$M$\EnE{} ¢¤ı “‰ã‰À ğ‰Ü‰Àı õ‰µ‰€‰‘û‰ü ¨‰´ ğ‰Â ø —‰€‰ú‰‘ ğ‰Â \InE{}$M$\EnE{} ª‰‘õ‰Û û‰ƒ‰º
  Ÿ‰‘¬‰Û›‰Ş‰â  õ‰Æ‰µ‰Ö‰ƒ‰İ ¥ —‰ã‰À¢ ÷‰‘õ‰µ‰€‰‘û‰ü ¥ş‰Âõ‰Àøñ ÷‰‘¬‰Ô‰Â ÷‰±‰‘ª‰À.

{\proof} ¤.í. [32].\InE{}$\blk$\EnE{}
{\pro.\label{g3-1}} ê‰Â­ ‰ƒ‰İ \InE{}$M$\EnE{} ş‰× \InE{}$-R$\EnE{}õ‰Àøñ “‰‘ª‰À. ¢¤ ş‰ß ¬‰¤– ª‰Âş‰Í ¥ş‰Â û‰İ ¤¥÷‰À:

\InE{}(i)\EnE{} \InE{}$G-dimM$\EnE{} õ‰µ‰€‰‘û‰ü ¨‰´.

 \InE{}(ii)\EnE{} ä‰À¢ ¬‰½‰ƒ‰¼ ø õ‰·‰±‰´ \InE{}$n$\EnE{} ø ¥ş‰Âõ‰Àøñû‰‘ı ş‰Ø‰€‰¡‰´ \InE{}$U_1 , U_2 ,\cdots, U_n$\EnE{}
 ø›‰¢ ¢¤÷‰À “‰‚ ω¤ı  \InE{}$\{U_1,U_2,\cdots ,U_n\}$\EnE{} ¡‰‘÷‰¢ùı õ‰Æ‰µ‰Ö‰Û ¥
 ¥ş‰Âõ‰Àøñû‰‘ı \InE{}$M$\EnE{} û‰Æ‰µ‰€‰À ø \InE{}$U_1\oplus U_2\oplus\cdots \oplus U_n\leq_e M$\EnE{} 

 \InE{}(iii)\EnE{} \InE{}$G-dimM=n$\EnE{}.
 
 \InE{}(iv)\EnE{} “‰Âı û‰Â ¥÷‰¹‰ƒ‰Â ê‰Ãş‰€‰Àù
 \InE{}$A_1\subseteq A_2\subseteq\cdots \subseteq A_n\subseteq\cdots $\EnE{} ¥ ¥ş‰Â
 õ‰Àøñû‰‘ı \InE{}$M$\EnE{} ä‰À¢ ¬‰½‰ƒ‰¼ ø õ‰·‰±‰´ \InE{}$j$\EnE{} ø›‰¢ ¢¤¢ “‰‚ ω¤ı  “‰Âı û‰Â 
 \InE{}$k\geq j$\EnE{}, \InE{}$A_j\subseteq_e A_k$\EnE{}.
 
\InE{}(v)\EnE{} \InE{}$M$\EnE{} ¢¤ ª‰Â¯\InE{}$ACC$\EnE{} ¤øı ¥ş‰Âõ‰Àøñû‰‘ı õ‰Ø‰Ş‰Û ¬‰Àë õ‰ü ‰À.

 \InE{}(vi)\EnE{} “‰Âı û‰Â ¥÷‰¹‰ƒ‰Â û‰€‰Àù
 \InE{}$A_1\supseteq A_2\supseteq\cdots \supseteq A_n\supseteq\cdots $\EnE{} ¥
 ¥ş‰Âõ‰Àøñû‰‘ı \InE{}$M$\EnE{} ä‰À¢ ¬‰½‰ƒ‰¼ ø õ‰·‰±‰´ \InE{}$j$\EnE{} ø›‰¢ ¢¤¢ “‰‚ ω¤ı  “‰Âı û‰Â \InE{}$k\geq j$\EnE{},
 \InE{}$A_k\subseteq_e A_j$\EnE{}.

{\proof} “‰Âı ™‰±‰‘– û‰İ ¤¥ “‰¢ö  \InE{}(i)\EnE{} —‰‘ \InE{}(v)\EnE{} “‰‚ \InE{}]\EnE{}72\InE{}[\EnE{} õ‰Â›‰ã‰‚ ª‰¢.
Ÿ‰‘ñ ™‰‘“‰´ õ‰ü ‰ƒ‰İ \InE{}(i)\EnE{} û‰İ ¤¥ \InE{}(vi)\EnE{} ¨‰´:
\InE{}(vi)$\Longrightarrow$(i)\EnE{} ¥ş‰Â ê‰Â­ õ‰ü ‰ƒ‰İ \InE{}$A'_1$\EnE{} õ‰Ø‰Ş‰Û \InE{}$A_1$\EnE{} “‰‘ª‰À ø
 \InE{}$A_1\cap A'_1=(0)$\EnE{}. “‰Àş‰ú‰ü ¨‰´ \InE{}$A_2\cap A'_1=(0)$\EnE{} ø \InE{}$A'_1$\EnE{} ¤ “‰‚ ¥ş‰Âõ‰Àøñ 
\InE{}$A'_2$\EnE{} —‰¨‰ã‰‚ õ‰ü¢û‰ƒ‰İ, ş‰ã‰€‰ü; õ‰Ø‰Ş‰Û \InE{}$A_2$\EnE{}. ¢¤ ÷‰µ‰ƒ‰¹‰‚ \InE{}$A_2\cap A'_2=(0)$\EnE{}.
ğ‰Â ş‰ß ¤ø© ¤ ¢õ‰‚ ¢û‰ƒ‰İ ş‰× ¥÷‰¹‰ƒ‰Â ¬‰ã‰¢ı ¥ ¥ş‰Âõ‰Àøñû‰‘ı õ‰Ø‰Ş‰Û ¢¤ş‰İ  ω±‰Õ 
\InE{}(v)\EnE{}, ä‰À¢ ¬‰½‰ƒ‰¼ ø õ‰·‰±‰´ \InE{}$j$\EnE{} ø›‰¢ ¢¤¢ “‰‚ ω¤ı  “‰Âı û‰Â \InE{}$k\geq j$\EnE{},
\InE{}$A'_j=A'_k$\EnE{}.Ÿ‰‘ñ ™‰‘“‰´ õ‰ü‰ƒ‰İ “‰Âı û‰Â  \InE{}$k\geq j$\EnE{}, \InE{}$A_k\subseteq_e A_j$\EnE{}.
ê‰Â­ ‰ƒ‰İ ‰€‰ƒ‰ß ÷‰±‰‘ª‰À. “‰€‰‘“‰Âş‰ß ¥ş‰Âõ‰Àøñ ÷‰‘¬‰Ô‰Â \InE{}$B_j$\EnE{} ¥ õ‰Àøñ \InE{}$A_j$\EnE{} ø›‰¢
 ¢¤¢ “‰‚ ω¤ı  \InE{}$B_j\cap A_k=(0)$\EnE{} “‰Âı ş‰× õ‰Ö‰À¤ \InE{}$k\geq j$\EnE{}.
 Ÿ‰‘ñ “‰Àş‰ú‰ü \InE{}$(A'_k+B_j)\cap A_k=(0)$\EnE{} .¢¤ ÷‰µ‰ƒ‰¹‰‚ \InE{}$A'_k=A'_k+B_j$\EnE{} ø
  \InE{}$B_j\subseteq A'_k=A_j$\EnE{} .“‰€‰‘“‰Âş‰ß
   \InE{}$B_j=B_j\cap A_j\subseteq A'_j\cap A_j=(0)$\EnE{} ø \InE{}$B_j=(0)$\EnE{}  —‰€‰‘ì‰Ë ¢¤ş‰İ.\\
\InE{}(i)$\Longrightarrow$(vi)\EnE{} ê‰Â­ ‰ƒ‰İ “‰ã‰À ğ‰Ü‰Àı \InE{}$M$\EnE{} õ‰µ‰€‰‘û‰ü ÷‰±‰‘ª‰À . “‰€‰‘“‰‚ ì‰Ì‰ƒ‰‚ \ref{t3-2} 
,\InE{}$\oplus_{i=1}^{\infty} N_i\subseteq M$\EnE{}. ¢¤ ÷‰µ‰ƒ‰¹‰‚
 \InE{}$\oplus_{i=1}^{\infty} N_i\supset \oplus_{i=2}^{\infty} N_i\supset\oplus_{i=3}^{\infty} N_i\supset\cdots$\EnE{} 
 ş‰ß ¥÷‰¹‰ƒ‰Â ¢¤ ª‰Â¯ \InE{}(vi)\EnE{} ¬‰Àë ÷‰Ş‰ü ‰À.\InE{}$\blk$\EnE{} 
{\pro.\label{g3-2}} ê‰Â­ ‰ƒ‰İ \InE{}$M$\EnE{} ş‰× \InE{}$-R$\EnE{}õ‰Àøñ “‰‘ª‰À \InE{}$G-dim M < \infty$\EnE{}. ¢¤ ş‰ß ¬‰¤– 

\InE{}(i)\EnE{} \InE{}$G-dim(M_1\oplus M_2)=G-dim(M_1)+G-dim(M_2)$\EnE{}.

\InE{}(ii)\EnE{} ğ‰Â \InE{}$A\subseteq M$\EnE{}, öğ‰‘ù\InE{}$G-dim(A)\leq G-dim(M)$\EnE{}. —‰Æ‰‘øı “‰Âì‰Â¤
 ¨‰´ ğ‰Â ø —‰€‰ú‰‘ ğ‰Â \InE{}$A\leq_e M$\EnE{}.

 \InE{}(iii)\EnE{} “‰Âı û‰Â ¥÷‰¹‰ƒ‰Â ê‰Ãş‰€‰Àù \InE{}$A_1\subseteq A_2\cdots\subseteq A_n\cdots$\EnE{}¥
 ¥ş‰Âõ‰Àøñû‰‘ı \InE{}$M$\EnE{} ä‰À¢ ¬‰½‰ƒ‰¼ ø õ‰·‰±‰´ \InE{}$k$\EnE{} ø›‰¢ ¢¤¢ “‰‚ ω¤ı  “‰Âı û‰Â
 \InE{}$i\geq k$\EnE{}, \InE{}$G-dim(A_i)=G-dim(A_k)$\EnE{} .

 \InE{}(iv)\EnE{} ğ‰Â\InE{}$A\leq M$\EnE{} ø  \InE{}$f:A\longrightarrow A$\EnE{} ş‰× —‰Ø‰Âş‰¿‰µ‰ü “‰‘ª‰À, öğ‰‘ù\InE{}$f(A)\leq_e A$\EnE{}.
 
{\proof} ¤.í. [52].\InE{}$\blk$\EnE{} 
{\pro.\label{g3-3}} ğ‰Â \InE{}$M$\EnE{} ş‰× \InE{}$-R$\EnE{}õ‰Àøñ “‰‘ “‰ã‰À ğ‰Ü‰Àı õ‰µ‰€‰‘û‰ü “‰‘ª‰À, öğ‰‘ù û‰Â ¥ş‰Âõ‰Àøñ ÷‰‘¬‰Ô‰Â
\InE{}$M$\EnE{} ª‰‘õ‰Û ¥ş‰Âõ‰Àøñ ş‰Ø‰€‰¡‰´ ¨‰´.

{\proof} ¤.í. [52] ¬‰Ô‰½‰‚ 01.\InE{}$\blk$\EnE{}