Bulletin of the
Korean Mathematical Society BKMS

A ring $R$ is called a UN ring if every non unit of it can be written as product of a unit and a nilpotent element. We obtain results about lifting of conjugate idempotents and unit regular elements modulo an ideal $I$ of a UN ring $R$. Matrix rings over UN rings are discussed and it is obtained that for a commutative ring $R$, a matrix ring $M_n(R)$ is UN if and only if $R$ is UN. Lastly, UN group rings are investigated and we obtain the conditions on a group $G$ and a field $K$ for the group algebra $KG$ to be UN. Then we extend the results obtained for $KG$ to the group ring $RG$ over a ring $R$ (which may not necessarily be a field).

Keywords: UN rings, units, nilpotents, group rings

MSC numbers: Primary 16N40, 16S34, 16U60, 20C05, 20C07