# Examples of stretched artinian local ring

In Sally's Paper stretched artinian local ring is defined as :

Let $$(R, \mathfrak{m})$$ be an Artin local ring of length $$\lambda.$$ If $$\nu$$ is the embedding dimension of $$R$$, that is, $$\nu$$ is the number of elements in a minimal basis of $$\mathfrak{m}$$, then $$\mathfrak{m}^{\lambda-\nu+1} = 0.$$ We will say that $$R$$ is stretched if $$\lambda - \nu$$ is the least integer $$i$$ such that $$\mathfrak{m}^{i+1} = 0.$$ If $$R$$ is not a field, $$R$$ is stretched if and only if $$\mathfrak{m}^2$$ is principal ideal.

$$\textbf{I am thinking how to construct some examples of stretched artinian local ring}.$$

$$\textbf{It will be better if could get those with \lambda \neq \nu and \mathfrak{m}^3 \neq 0}.$$

Any help is highly appreciated.

Some of the easy examples are $$k[x,y]/(x^2,xy,y^n)$$.
Try $$k[x_0,x_1,\ldots,x_{2n}]/(R)$$ where $$R$$ consists of the relations $$x_0^{\lambda-2n-1}=x_1x_2=x_3x_4=\cdots=x_{2n-1}x_{2n},$$ together with $$x_ix_j=0$$ for the remaining products and squares (except $$x_0^2$$ of course) - which imply that $$x_0^{\lambda-2n}=0$$. This has length $$\lambda$$, $$\nu=2n+1$$, and the $$(\lambda-2n)$$th power of $$\mathfrak{m}$$ is the first to be zero. This is even a Gorenstein ring. There are many obvious variations on this example.