2

Let $M$ be an $R$-module, $M$ is said to be uniserial if submodules of $M$ are totally ordered by inclusion. That is if $N$ and $L$ are two submodules of $M$ the either $N\subseteq L$ or $L\subseteq N$.

My question is can we drop equality here like just we can say either $N\subsetneq L$ or $L\subsetneq N$ for any two submodules of $M$. Or equality is necessary!

I found this definition on Wikipedia.

sushil soni
  • 335

1 Answers1

4

Equality is necessary in the sense that the definition is silly without it. Specifically, if your definition is, "if $N$ and $L$ are two submodules of $M$, then either $N \subsetneq L$ or $L \subsetneq N$", then nothing is uniserial, since when $N=L=M$ we have neither proper containment.

Joshua P. Swanson
  • 9,634