Local rings of minimal length
Let \((R,\mathfrak m)\) denote a local ring. A regular fractional ideal \(\omega\) of \(R\) is called an \(\mathfrak m\)-canonical ideal provided \(\omega : (\omega : I) = I\) for any regular fractionary ideal \(I\) of \(R.\) Assume that \(R\) possesses an \(\mathfrak m\)-canonical ideal \(\omega,\) and let \(t\) denote its minimal number of generators. Then \(l_R(\omega/R) \geq t-1,\) and \(R\) is called of minimal length if equality holds. The author considered the following conditions: (1) \(R\) is of minimal length. (2) Each regular ideal of \((\mathfrak m : \mathfrak m)\) is divisorial as ideal of \(R.\) (3) \(\mathfrak m\) is an \(\mathfrak m\)-canonical ideal of \((\mathfrak m : \mathfrak m).\) Then it is shown that (1)\(\Longrightarrow\)(2)\(\Longleftrightarrow\)(3). Moreover, all the conditions are equivalent for \(R\) a one-dimensional analytically unramified and residually rational ring. As illustrations of the results several examples are considered.
scientific article; zbMATH DE number 5534968
scientific article