반단순 가군

환론에서 반단순 가군(半單純加群, 영어: semisimple module)은 단순 가군들의 직합으로 분해되는 가군이다.

정의

환 $R$ 위의 왼쪽 가군 $_{R}M$ 에 대하여, 다음 조건들이 서로 동치이며, 이를 만족시키는 가군을 왼쪽 반단순 가군이라고 한다.

왼쪽 단순 가군들의 직합으로 나타낼 수 있다. 즉, $_{R}M\cong \bigoplus _{i\in I}{}_{R}N_{i}$ 가 되는 단순 가군의 집합 $\{{}_{R}N_{i}\}_{i\in I}$ 이 존재한다.
$_{R}M$ 의 모든 단순 부분 가군들의 합이다. 즉, $_{R}M$ 의 단순 부분 가군들이 $\{{}_{R}N_{i}\}_{i\in I}$ 일 때, $\textstyle {}_{R}N=\sum _{i\in I}{}_{R}N_{i}$ 로 정의한다면, ${}_{R}N={}_{R}M$ 이다. (여기서 $\textstyle \sum _{i}$ 은 $M$ 의 원소의 합이다.)
임의의 부분 가군 $_{R}N\subseteq {}_{R}M$ 에 대하여, $_{R}M\cong {}_{R}N\oplus {}_{R}P$ 인 왼쪽 가군 ${}_{R}P$ 가 존재한다.
$\operatorname {soc} {}_{R}M={}_{R}M$ 이다. 여기서 $\operatorname {soc}$ 는 가군의 주각이다.

오른쪽 반단순 가군도 마찬가지로 정의된다.

성질

반단순 가군의 부분 가군과 몫가군 역시 반단순 가군이다.

반단순 가군들의 직합 역시 반단순 가군이다.

환 $R$ 위의 반단순 가군 $M$ 의 자기준동형환 $\operatorname {End} _{R}M$ 은 폰 노이만 정칙환이다 (따라서 반원시환이다).

반단순환 위의 모든 가군은 반단순 가군이다.

예

나눗셈환 $D$ 위의 모든 가군은 반단순 가군이다. (이는 나눗셈환은 반단순환이기 때문이다.) 이 경우, 단순 가군은 나눗셈환 $D$ 스스로이며, 모든 자유 가군은 $D$ 의 직합과 동형이다. 특히, 체 위의 모든 벡터 공간은 반단순 가군이다.

정수환 $\mathbb {Z}$ 위의 가군은 아벨 군이며, 정수환 위의 단순 가군은 (아벨 군인 단순군이므로) 소수 크기의 순환군 $\operatorname {Cyc} (p)$ 이다. 따라서, 정수환 위의 반단순 가군은 소수 크기의 순환군들의 직합이다.

같이 보기

반단순환

외부 링크

“Completely-reducible module”. 《Encyclopedia of Mathematics》 (영어). Springer-Verlag. 2001. ISBN 978-1-55608-010-4.
“Semi-simple module”. 《Encyclopedia of Mathematics》 (영어). Springer-Verlag. 2001. ISBN 978-1-55608-010-4.
“Semisimple rings and modules”. 《Mathematics and Such》 (영어). 2014년 12월 30일.

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