Let $(A,m)$ be a local commutative associative algebra over the field of complex numbers, $m^n\ne 0$, $m^{n+1}=0$ for some $n>0$, and


(1) $A$ is finite dimensional as vector space


(2) for any nonzero ideal $I$ of $A$, we have $m^n\subset I$


What can we say about such an $A$? For example, whether it is always a quotient algebra of some $\mathbb C[t^{m\_1},\ldots,t^{m\_k}]$? 


I wonder whether there are some results charactered this kind of algebra more explicitly



A problem for finite dimensional commutative algebra
Such a ring has simple socle, namely $\mathfrak m^n$. It follows easily that $A$ is self-injective, Gorenstein and, of course, of dimension $0$. You can construct them using Macaulay's method of *inverse system*; this is explained in Eisenbud's book on commutative algebra, if I recall correctly.


If $A\_4=\mathbb C[x,y]/(x^2,y^2)$, a four-dimensional example, then you cannot embed it in an algebra of the form $\mathbb C[t]/(t^\ell)$.


