We say that an ideal $\mathfrak a$ of $A$ is finitely generated if $\mathfrak a =(x_1,\cdots,x_n)=\sum_{i=1}^{n} Ax_i$, i.e. finitely generated as an $A$-module.
Is there a name for when $\mathfrak a$ is generated by all the finite products of the $x_i$? In other words, every element of $\mathfrak a$ 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 $\mathfrak a$ is not a ring and does not contain the constant terms.
Best Answer
If $\alpha$ is generated by all the finite products of the $x_i$, then it is generated by the $x_i$. In other words, $$(x_1,x_2,\ldots,x_n)=(x_1,x_2,\ldots,x_n,x_1^2,x_1x_2,\ldots,x_n^2,\ldots).$$ So there is not a separate concept of an ideal being finitely generated like there is for $A$-modules vs. $A$-algebras.