Rings with ideal-symmetric ideals
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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.
Abstract
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.
참고문헌 (12)
- D. Anderson, "Semigroups and rings whose zero products commute", Comm. Algebra, 27(6), pp.2847~2852 (1999)
- G. Birkenmeier, "Polynomial extensions of Baer and quasi-Baer rings", J. Pure Appl. Algebra, 159, pp.25~42 (1999)
- V. Camillo, "Ideal-symmetric and semiprime rings", Comm. Algebra, 41(12), pp.4504~4519 (2013)
- J. Clark, "Twisted matrix units semigroup algebras", Duke Math. J, 34, pp.417~424 (1967)
- P. Cohn, "Reversible rings", Bull. London Math. Soc, 31(6), pp.641~648 (1999)
- Y. Hirano, "On annihilator ideals of a polynomial ring over a noncommutative ring", J. Pure Appl. Algebra, 168(1), pp.45~52 (2002)
- C. Hong, "Ore extensions of quasi-Baer rings", Comm. Algebra, 37(6), pp.2030~2039 (2009)
- I. Kaplansky, "Rings of Operators, Math. Lecture Note Series" (1965), Benjamin
- T. Kwak, "Re exive property of rings", Comm. Algebra, 40(4), pp.1576~1594 (2012)
- J. Lambek, "On the representation of modules by sheaves of factor modules", Canad. Math. Bull, 14, pp.359~368 (1971)
- G. Marks, "Reversible and symmetric rings", J. Pure Appl. Algebra, 174(3), pp.311~318 (2002)
- G. Mason, "Re exive ideals", Comm. Algebra, 9(17), pp.1709~1724 (1981)