The socle is always an ideal. Let c generate a minimal left ideal, and suppose cx does not generate a minimal left ideal, so that for some y, ycx does not generate cx. It follows that yc cannot generate c, which contradicts Rc a minimal left ideal. Therefore cx is in the socle, and the socle is an ideal.
If R is simple and left artinian, i.e. R is the n×n matrices over a division ring, it has a nonzero socle that is all of R. Remember that a minimal left ideal is a single column, or perhaps as many as n columns where each is a prescribed multiple of the other. Since each column is a minimal ideal, combine them and get all of R. Similarly, the right socle is the combination of all n individual rows, and is R.
Although they are both ideals, the left socle need not equal the right socle. Let R be the matrix ring [F,F|0,K], where F/K is a field extension. The left socle is [F,F|0,0], while the right socle is [0,F|0,K].
Recall that c generates a minimal left ideal iff it generates a minimal right ideal in a semiprime ring. Thus the left and right socles are the same in a semiprime ring, or any ring that implies a semiprime ring.