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Keywords:
$\sqrt {J}U$ ring; $JU$ ring; $UU$ ring; Jacobson radical; nilpotent
$\sqrt {J}U$ ring; $JU$ ring; $UU$ ring; Jacobson radical; nilpotent
Summary:
We intend to unviel a new class of rings, called $\sqrt {J}U$ rings, if the units of a ring $R$ equal the sum of 1 and an element from $\sqrt {J(R)}$. Recall that $\sqrt {J(R)}$ is a subset of $R$, not necessarily a subring, which equals $\{ z\in R \colon z^n \in R$ for some $n \geq 1 \}$. Both $UU$ and $JU$ rings are $\sqrt {J}U$ rings credited to the fact that nilpotents and $J(R)$ are subsets of $\sqrt {J(R)}$. The properties exhibited by $\sqrt {J}U$ rings are explored in a thorough manner following which its relations with other rings are observed. For instance, $UNJ$ rings are $\sqrt {J}U$ and no matrix ring, when $n > 1$ is $\sqrt {J}U$. We have focused on extensions of $\sqrt {J}U$ rings like $T(R,M)$, $H_{(p,q)}(R)$, $L_{(p,q)}(R)$, Morita context and group rings.
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