Kurosh–Amitsur radical theory

Definition

Work with associative rings that need not have an identity. A class $\mathcal{R}$ of such rings, closed under isomorphism, is a radical class in the sense of Kurosh and Amitsur if the following hold.

1. If $A \in \mathcal{R}$ and $B$ is a homomorphic image of $A$, then $B \in \mathcal{R}$.
2. Every ring $A$ has a two-sided ideal $\mathcal{R}(A)$ that belongs to $\mathcal{R}$ and contains every ideal of $A$ that belongs to $\mathcal{R}$.
3. For every ring $A$, the quotient $A/\mathcal{R}(A)$ has no nonzero ideal in $\mathcal{R}$, that is, $\mathcal{R}(A/\mathcal{R}(A)) = 0$.

The ideal $\mathcal{R}(A)$ is the $\mathcal{R}$-radical of $A$. A ring $A$ is $\mathcal{R}$-semisimple if $\mathcal{R}(A) = 0$.