I am thinking of the question that when the (right) socle of a ring $R$ (as a right $R$-module) is small? In fact, since the Jacobson radical $J(R)$ of $R$ is the largest small right ideal, my question asks when is the socle of $R$ (the sum of all minimal right ideals of $R$) a subset of $J(R)$? Thanks in advance for any suggestion or answer!
2026-03-30 04:44:10.1774845850
When is the socle small?
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I'm going to assume you know that the Jacobson radical is the set of elements annihilating all simple right $R$-modules.
One obvious thing is that if $soc(R_R)$ is small, then $soc(R_R)^2=\{0\}$. On the other hand, the Jacobson radical contains nilpotent ideals, so the converse holds as well.
This gives a pretty simple criterion for determining if a ring has a small socle or not.