I started studying the idea of semi-simple and I wanted to double-check why this composition makes sense but before that
Def: A ring with $1_R$ is said to be semi-simple if the free left $R$-module underlying $R$ is a (direct) sum of simple $R$- modules.
The decomposition of a semi-simple ring is given as follows
$$A=\bigoplus_{i=1}^n\bigoplus_{j=1}^{k_i} L_{i,j}$$ where each $L_i$ is a simple left ideal and $A$ is any ring.
My Uncertainties:
To start of, I wanted to make sure my understanding on a semi-simple ring is correct. Would I be correct in thinking that a semi-simple ring $A$ is basically a semi-simple module where the ring $A$ is viewed as a $A$-module? Then adopting the ideas from modules, this is basically saying any semi-simple ring $A$ can be decomposed into simple $A$-submodules, which in this context, are just simple left ideals and so the composition makes sense?
A Question:
Why does the definition that I saw online stressed about the left $R$-module being free? Would the existence of the identity element automatically guarantees the module to be free because we can multiply everything by $1_R$ to get everything back?
Is my understanding correct? Please let me know and many thanks in advance!
To say that $_AA$ is a semisimple module, or that $A_A$ is a semsimple module, is equivalent to saying that $R$ is a semisimple (Artinian) ring, yes.
Possibly it is just a redundancy to make sure the reader knows they are talking about the regular module $_RR$. The freeness is immaterial to the definition.
It is extremely uncommon (and unproductive, at the beginning) to consider the theory of semisimple rings and modules without identity. My guess is your book wants you to assume identity.