A ring is semiprimitive if and only if it has a faithful semisimple left module.
A commutative ring is semiprimitive if and only if it is a subdirect product of fields, (Lam 1995, p. 137).
A left artinian ring is semiprimitive if and only if it is semisimple, (Lam 2001, p. 54).
Such rings are sometimes called semisimple Artinian, (Kelarev 2002, p. 13).
However, this is a stricter notion, since the endomorphism ring of a countably infinite dimensional vector space is semiprimitive, but not a subdirect product of simple rings, (Lam 1995, p. 42).