What exactly is a trivial module?

Yes, this is a quite basic answer, but I have to admit to be absolutely confused about this notion.

Searching on the web, I managed to found two possible definition of trivial modules, referring actually to two different mathematical objects.

The first one is just the singleton set with the only possible module structure, also called the zero module.

The second one, that is the definition that is leading me to be confused and a bit stuck after so much hours of studying, is the following:

Let $A$ be a ring, $M$ an abelian group. $M$ is called a trivial module if it is a module endowed with the trivial action.

But...what exactly is a trivial action? Yes, of course the first things that I think is the trivial $ax=x$ for each $x \in M$, but there is something wrong with it, because directly from the axioms of modules I have:

for each $x \in M, (1+1)x = x$ (because the action is trivial), and $(1+1)x = 1x+1x=x+x$ that implies $x = 0$, i.e. $M$ is the group $0$.

Please, could you help me in understanding what am I missing? Thank you very much!!!

Ps: this question is related to this one (From $G$-mod to $\mathbb{Z}G$-mod and a related question.) where I believed to have understood this definition :)

Solution 1:

Trivial action of ring $A$ on Abelian group $M$:

$am=0$ for all $a\in A$ and $m\in M$.

If $A$ has an identity element (and the axiom $1x=x$ is posed), then this forces $m=1m=0$ for all $m$, hence $M=\{0\}$.

Solution 2:

You seem to be just confusing different notions of module.

• $G$-module would be an abelian group $M$ with the action of a group $G$ compatible with addition. (This can well be the trivial action.)

• $R$-module, or just module, where $M$ is an abelian group and one has scalar multiplication with the elements from the ring $R$ similarly to case of vector-spaces. (This can also be understood as an instance of the former.)

To make matter worse it is also not uncommon to have an $R$-module with an additional group action.