I know the following theorems:
(1) Let $D$ be a skew field and $R= M_n(D)$. Then $R$ is simple, left semisimple, left Artinian and left Noetherian. Moreover, $R$ has a unique (up to isomorphism) simple left faithful submodule $M$ such that $R \cong M \oplus \dots \oplus M$ where $R$ is considered to be the regular left $R$-module.
(2) The finite direct product of left semisimple rings is left semisimple.
(3) Let $R$ be a left semisimple ring. Then
$$R \cong M_{n_1}(D_1) \times \dots M_{n_k}(D_k)$$
for skew fields $D_1, \dots, D_k$ and $n_1, \dots, n_k \geq 1$. The number $k$ and the pairs $(n_i, D_i)$ are uniquely determined (up to permutation).
Do these theorems have a right version? I.e. if I replace the word left to right in these theorems, are these theorems true? I guess the answer is an obvious yes, but I want a quick sanity check. The same proofs with modifications seem to apply.
I'm asking this because I am asked to prove that a ring $R$ is left semisimple if and only if the ring is right semisimple.
If the answer to my first question is yes, then we can argue as follows:
Assume $R$ is left semisimple. Then by $(3)$ $R$ is a product of matrix rings over skew fields. By the right versions of $(1),(2)$, $R$ is also right semisimple.
Conversely is done in the same way: if $R$ is right semisimple, then by the right version of $(3)$ $R$ is a product of matrix rings over skew fields. By $(1)$ and $(2)$, $R$ is left semisimple.
Is all of the above correct?
Yes, everything works if you replace "left" with "right". In fact, instead of modifying the proofs, you can deduce the right cases from the left cases as follows. Given a ring $R$, let $R^{op}$ be $R$ but with the order of multiplication flipped. Note that a right $R$-module is the same thing as a left $R^{op}$-module. So, $R$ is right semisimple, Noetherian, etc. iff $R^{op}$ is left semisimple, Noetherian, etc. Moreover, if $D$ is a skew field, then so is $D^{op}$, and $M_n(D)^{op}\cong M_n(D^{op})$ by the transpose map.
So, if you apply the theorems to $R^{op}$ instead of to $R$ (and similarly with all the skew fields), you just get the theorems for $R$ but with "left" replaced by "right" everywhere.