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Consider a commutative noetherian local ring $R$ of dimension $d$ and define

$$c_R\colon=\min_{(x_1,\ldots,x_d)} \{\mathrm{length}\ R/(x_1,\ldots,x_d)R\mid (x_1,\ldots,x_d)\ \mathrm{is\ a\ system\ of parameters\ of\ }R\}.$$

1. Does $c_R$ always give information about the ring $R$? For instance, clearly $c_R=1$ if and only if $R$ is regular. When is $c_R=2$? Does $R$ have to be any particular type of ring for $c_R$ to be $2$?

2. In a given local ring $R$ can one characterize those system of parameters $(x_1,\ldots,x_d)$ for which $c_R=\mathrm{length}\ R/(x_1,\ldots,x_d)R$?

Consider a commutative noetherian local ring $R$ of dimension $d$ and define

$$c_R\colon=\min_{(x_1,\ldots,x_d)} \{\mathrm{length}\ R/(x_1,\ldots,x_d)R\mid (x_1,\ldots,x_d)\ \mathrm{is\ a\ system\ of parameters\ of\ }R\}.$$

1. Does $c_R$ always give information about the ring $R$? For instance, clearly $c_R=1$ if and only if $R$ is regular. When is $c_R=2$? Does $R$ have to be any particular type of ring for $c_R$ to be $2$?

2. In a given local ring $R$ can one characterize those system of parameters $(x_1,\ldots,x_d)$ for which $c_R=\mathrm{length}\ R/(x_1,\ldots,x_d)R$? For instance, one could say that a necessary condition for $c_R=\mathrm{length}\ R/(x_1,\ldots,x_d)R$ is that $(x_1,\ldots,x_d)$ has to be "maximal", that is, not contained in any larger parameter ideal. Is this condition also sufficient?

Consider a commutative noetherian local ring $R$ of dimension $d$ and define

$$c_R\colon=\min_{(x_1,\ldots,x_d)} \{\mathrm{length}\ R/(x_1,\ldots,x_d)R\mid (x_1,\ldots,x_d)\ \mathrm{is\ a\ system\ of parameters\ of\ }R\}.$$

1. Does $c_R$ always give information about the ring $R$? For instance, clearly $c_R=1$ if and only if $R$ is regular. When is $c_R=2$? Does $R$ have to be any particular ring for $c_R$ to be $2$?

2. In a given local ring $R$ can one characterize those system of parameters $(x_1,\ldots,x_d)$ for which $c_R=\mathrm{length}\ R/(x_1,\ldots,x_d)R$?

Consider a commutative noetherian local ring $R$ of dimension $d$ and define

$$c_R\colon=\min_{(x_1,\ldots,x_d)} \{\mathrm{length}\ R/(x_1,\ldots,x_d)R\mid (x_1,\ldots,x_d)\ \mathrm{is\ a\ system\ of parameters\ of\ }R\}.$$

1. Does $c_R$ always give information about the ring $R$? For instance, clearly $c_R=1$ if and only if $R$ is regular. When is $c_R=2$? Does $R$ have to be any particular type of ring for $c_R$ to be $2$?

2. In a given local ring $R$ can one characterize those system of parameters $(x_1,\ldots,x_d)$ for which $c_R=\mathrm{length}\ R/(x_1,\ldots,x_d)R$?

Consider a commutative noetherian local ring $R$ of dimension $d$ and define

$$c_R\colon=\min_{x_1,\ldots,x_d} \{\mathrm{length}\ R/(x_1,\ldots,x_d)R\mid (x_1,\ldots,x_d)\ \mathrm{is\ a\ system\ of parameters\ of\ }R\}.$$

1. Does $c_R$ always give information about the ring $R$? For instance, clearly $c_R=1$ if and only if $R$ is regular. When is $c_R=2$? Does $R$ have to be any particular ring for $c_R$ to be $2$?

2. In a given local ring $R$ can one characterize those system of parameters $(x_1,\ldots,x_d)$ for which $c_R=\mathrm{length}\ R/(x_1,\ldots,x_d)R$?

Consider a commutative noetherian local ring $R$ of dimension $d$ and define

$$c_R\colon=\min_{(x_1,\ldots,x_d)} \{\mathrm{length}\ R/(x_1,\ldots,x_d)R\mid (x_1,\ldots,x_d)\ \mathrm{is\ a\ system\ of parameters\ of\ }R\}.$$

1. Does $c_R$ always give information about the ring $R$? For instance, clearly $c_R=1$ if and only if $R$ is regular. When is $c_R=2$? Does $R$ have to be any particular ring for $c_R$ to be $2$?

2. In a given local ring $R$ can one characterize those system of parameters $(x_1,\ldots,x_d)$ for which $c_R=\mathrm{length}\ R/(x_1,\ldots,x_d)R$?