Rings whose ideals are isomorphic to trace ideals

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  • Toshinori Kobayashi
    Graduate School of Mathematics Nagoya University Furocho, Chikusaku Nagoya Aichi 464‐8602 Japan
  • Ryo Takahashi
    Graduate School of Mathematics Nagoya University Furocho, Chikusaku Nagoya Aichi 464‐8602 Japan

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<jats:title>Abstract</jats:title><jats:p>Let <jats:italic>R</jats:italic> be a commutative noetherian ring. Lindo and Pande have recently posed the question asking when every ideal of <jats:italic>R</jats:italic> is isomorphic to some trace ideal of <jats:italic>R</jats:italic>. This paper studies this question and gives several answers. In particular, a complete answer is given in the case where <jats:italic>R</jats:italic> is local: it is proved in this paper that every ideal of <jats:italic>R</jats:italic> is isomorphic to a trace ideal if and only if <jats:italic>R</jats:italic> is an artinian Gorenstein ring, or a 1‐dimensional hypersurface with multiplicity at most 2, or a unique factorization domain.</jats:p>

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