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Let $(R, \mathfrak m)$ be a local Cohen-Macaulay ring. How to show that $R$ has minimal multiplicity (i.e. $e(R)=\mu (\mathfrak m)- \dim R +1$ ) if and only if $\mathfrak m^2=(\overline x)\mathfrak m$ for some regular sequence $\overline x$ in $R$ ?

If this is not true in general is it at least true if we assume $R/\mathfrak m$ is infinite ?

Let $(R, \mathfrak m)$ be a local Cohen-Macaulay ring. How to show that $R$ has minimal multiplicity (i.e. $e(R)=\mu (\mathfrak m)- \dim R +1$ ) if and only if $\mathfrak m^2=(\overline x)\mathfrak m$ for some regular sequence $\overline x$ in $R$ ?

Let $(R, \mathfrak m)$ be a local Cohen-Macaulay ring. How to show that $R$ has minimal multiplicity (i.e. $e(R)=\mu (\mathfrak m)- \dim R +1$ ) if and only if $\mathfrak m^2=(\overline x)\mathfrak m$ for some regular sequence $\overline x$ in $R$ ?

If this is not true in general is it at least true if we assume $R/\mathfrak m$ is infinite ?

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Local Cohen-Macaulay ring of minimal multiplicity

Let $(R, \mathfrak m)$ be a local Cohen-Macaulay ring. How to show that $R$ has minimal multiplicity (i.e. $e(R)=\mu (\mathfrak m)- \dim R +1$ ) if and only if $\mathfrak m^2=(\overline x)\mathfrak m$ for some regular sequence $\overline x$ in $R$ ?