Note that Zp2 is a semisimple Z module, while Zp2 is not.
The ring R is semisimple if it is a semisimple left R module. The submodules of the ring Z are the various multiples of n, and none of these are summands, hence Z is not a semisimple ring.
Now for a confusing definition. The ring R is simple if it has exactly two (two-sided) ideals, 0 and itself. Note that R may be a simple ring, but not a simple R module, if it has intermediate left ideals.
A simple or semisimple module inherits the adjective "left" or "right" from the module itself, usualy by context, and similarly for a semisimple ring, but a simple ring always refers to two-sided ideals. Even worse, one can say things like module and semisimple, with left or right understood by context, but one must always say left/right ideal, else a two-sided ideal is assumed. This is the terminology that has evolved for noncommutative algebra.
The precedent set by noetherian rings doesn't apply either. a noetherian ring is both left noetherian and right noetherian, but a simple ring need not be left simple or right simple. If we happen to run into a ring that is a simple left module and a simple right module simultaneously, I guess I'll call it left and right simple.
I may, from time to time, use terms like left simple or left semisimple, when asymmetric modules/ideals are involved. This might make things a little clearer.
If R is simple and commutative, its maximal ideal must equal 0, hence it is a field.
Every division ring is simple, and left and right simple.
If R is a left simple ring then any nonzero x generates all of R, and is left invertible. This makes R a division ring. Such a ring is also right simple. Perhaps that is why we don't spend much time worrying about left simple or right simple rings. They are just division rings.