Bulletin of the
Korean Mathematical Society BKMS

An ideal $I$ of a ring $R$ is strongly $\pi$-regular if for any $x\in I$ there exist $n\in {\mathbb N}$ and $y\in I$ such that $x^n=x^{n+1}y$. We prove that every strongly $\pi$-regular ideal of a ring is a $B$-ideal. An ideal $I$ is periodic provided that for any $x\in I$ there exist two distinct $m,n\in {\mathbb N}$ such that $x^m=x^n$. Furthermore, we prove that an ideal $I$ of a ring $R$ is periodic if and only if $I$ is strongly $\pi$-regular and for any $u\in U(I)$, $u^{-1}\in {\mathbb Z}[u]$.

Keywords: strongly $\pi$-regular ideal, $B$-ideal, periodic ideal

MSC numbers: 16E50, 16U50, 16E20