Semiprimary ring

Remarks

Example: the ring $R$ of upper triangular $2 \times 2$ matrices over a field $k$ is semiprimary: $J$ is the set of strictly upper triangular matrices, $J^2 = 0$, and $R/J \cong k \times k$ is semisimple. Non-example: $\Z_{(p)}$ satisfies clause 1, since $\Z_{(p)}/p\Z_{(p)} \cong \mathbb{F}_p$, but not clause 2, since $p^n \ne 0$ for every $n$. Every left Artinian ring is semiprimary, and a semiprimary ring is left Artinian if and only if it is left Noetherian. The definition is left-right symmetric.