Recall that a ring is left semisimple iff it is right semisimple, and J is a two sided ideal, so we don't have to talk about left or right semiprimary. The ring is either semiprimary or it is not.
A semisimple ring is jacobson semisimple, and is trivially semiprimary.
Let R be left artinian with jacobson radical J. Recall that J is nilpotent. Now R/J is jacobson semisimple, and is left artinian courtesy of R, hence R/J is semisimple. This makes R semiprimary. Every artinian ring is semiprimary.
If you want a semiprimary ring that is not left artinian, let R be right artinian, but not left artinian. It is still semiprimary, but not left artinian.