Semiprimitive ring

Remarks

Example: $\Z$ is semiprimitive, since its maximal ideals are $p\Z$ for primes $p$ and $\bigcap_p p\Z = 0$. Non-example: the local ring $\Z_{(p)}$ of fractions with denominator prime to $p$ is not semiprimitive, since $J(\Z_{(p)}) = p\Z_{(p)} \ne 0$. Equivalently, $R$ is a subdirect product of left primitive rings, or $R$ has a faithful semisimple left module. The definition is left-right symmetric, because $J(R)$ is also the intersection of the maximal right ideals. Other names are Jacobson semisimple and J-semisimple.