Bulletin of the
Korean Mathematical Society BKMS

Let $R$ be a ring with identity. An ideal $N$ of $R$ is called $ideal$-$symmetric$ (resp., $ideal$-$reversible$) if $ABC \subseteq N$ implies $ACB \subseteq N$ (resp., $AB \subseteq N$ implies $BA \subseteq N$) for any ideals $A, B, C$ in $R$. A ring $R$ is called $ideal$-$symmetric$ if zero ideal of $R$ is ideal-symmetric. Let $S(R)$ (called the $ideal$-$symmetric$ $radical$ of $R$) be the intersection of all ideal-symmetric ideals of $R$. In this paper, the following are investigated: (1) Some equivalent conditions on an ideal-symmetric ideal of a ring are obtained; (2) Ideal-symmetric property is Morita invariant; (3) For any ring $R$, we have $S(M_{n}(R)) = M_{n}(S(R))$ where $M_{n}(R)$ is the ring of all $n$ by $n$ matrices over $R$; (4) For a quasi-Baer ring $R$, $R$ is semiprime if and only if $R$ is ideal-symmetric if and only if $R$ is ideal-reversible.

Keywords: symmetric ideal, ideal-symmetric ideal, ideal-reversible ideal, ideal-symmetric ring, ideal-reversible ring, Morita invariant, ideal-symmetric radical, Baer ring, quasi-Baer ring

MSC numbers: 16D25, 16S50