Journal of the
Korean Mathematical Society JKMS

We focus our attention on a ring property that nilpotents are contained in the Jacobson radical. This property is satisfied by NI and left (right) quasi-duo rings. A ring is said to be $\it NJ$ if it satisfies such property. We prove the following: (i) Kothe's conjecture holds if and only if the polynomial ring over an NI ring is NJ; (ii) If $R$ is an NJ ring, then $R$ is exchange if and only if it is clean; and (iii) A ring $R$ is NJ if and only if so is every (one-sided) corner ring of $R$.

Keywords: NJ ring, Kothe's conjecture, Jacobson radical, NI ring, right quasi-duo ring, nilradical, (skew) polynomial ring

MSC numbers: 16N20, 16N40