What is meant by a regular $FG$-module. $G$ is a group and I believe $F$ is supposed to be a field. I'm completely confused by this concept on a question sheet and I can find lots of uses of the notation on the internet but not a proper definition. The example I'm struggling with now is the following:
Let $G$ be the cyclic group of order 3. Write the regular $\mathbb{R}G$-module as the direct sum of irreducible submodules.
If someone could explicitly define the regular $\mathbb{R}G$-module then that would be very helpful.
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$\begingroup$The regular representation of a group $G$ over a field $F$ is just the module ${}_{FG}(FG)$ which is over the $F$-algebra $FG$ that consists of (finite) formal sums $\sum_i f_ig_i$ with $f_i\in F,\,g_i\in G$, that is a vector space with basis $G$ and with multiplication inherited from $G$.
An $FG$-module is said to be irreducible (as representation of $G$), if it is simple as $FG$-module: has no nontrivial submodules.
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