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
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 $\mathbb C[t]/(t^\ell)$.
If $A$ is an Artin local ring of the form you have, we may write it as the quotient of a power series ring. Now, by Flenner's Bertini theorem (I believe it applies here), you have a one-dimensional domain quotient $B$ of the power series ring and $A$ is a quotient of $B$. The integral closure of $B$ is the power series ring in one variable. So, $A$ is the subquotient of $\mathbb{C}[[t]]$.
