We say that an ideal $\alpha$ of $A$ is finitely generated if $\alpha =(x_1,\cdots,x_n)=\sum_{i=1}^{n} Ax_i$, i.e. finitely generated as an $A$-module.
Then how we call if $\alpha$ is generated by all the finite products of the $x_i$? In other words, every element of $\alpha$ is a polynomial in $A[x_1,\cdots,x_n]$ with no constant term. It is similar to the finitely generated $A$-algebra, but it is not an $A$-algebra since $\alpha$ is not a ring and does not contain constant term.