Product of ideal generators

Question

Can we in general say that if we have an ideal $(I,J)$ that this is the same as the ideal $(I,J,IJ)$, where $IJ$ is the product of I and J?

Answer 1

Yes, they are the same. An ideal is closed under addition (i.e. the sum of two elements of an ideal is also in the ideal) and under multiplication with some element of the ring (i.e. the product of some element of the ideal and some element of the ring is also in the ideal). The product of two ideals is defined by $IJ=\{\sum_{i=1}^na_ib_i\mid n\in\mathbb{N}\land\forall i\in\{1,\ldots,n\}:(a_i\in I\land b_i\in J)\}$. Thus $IJ$ is a subset of $I\cap J$ and in particular a subset if $I$. Thus $(I,J,IJ)$ is contained in $(I,J)$ which again is certainly contained in $(I,J)$.