# Finitely generated ideal

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.

• E.g., $Ax^2\subseteq Ax$. Commented Jun 14, 2011 at 1:38
• @Jonas: I'm not sure I understand your comment. Could you expand a bit? Commented Jun 14, 2011 at 1:40
• @Zev: Sorry I was unclear. As a consequence of that containment, the ideal generated by $x\in A$ is the same as that generated by $\{x,x^2\}$, and so on for higher powers. Similarly, $Ax_ix_j\subseteq Ax_j$, etc. I didn't mean to be cryptic, but I was brief because you had already answered the question. Commented Jun 14, 2011 at 1:43

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.