Product of ideals

Definition

Definition with symbols

Suppose $I,J$ are ideals in a commutative unital ring $R$. Then the product of ideals $I$ and $J$, denoted $IJ$, is defined in the following equivalent ways:

• It is the additive subgroup generated by all elements of the form $ab$ where $a \in I, b \in J$
• It is the smallest ideal containing all elements of the form $ab$ where $a \in I, b \in J$
• It is the ideal defined as the set of elements of the form:

$\sum_{i=1}^n a_ib_i$

with $a_i \in I, b_i \in J$

Facts

• Product of ideals is commutative and associative. Hence, we can talk of the product of more than two ideals by simply writing them as a string. The product of ideals $I_1, I_2, \ldots, I_n$, denoted $I_1I_2\ldots I_n$, is the subgroup generated by elements of the form $a_1a_2\ldots a_n$ where $a_j \in I_j$ for every $j$
• We can also use this to define the notion of power of an ideal. For an ideal $I$, the ideal $I^n$ is simply $II \ldots I$ written $n$ times. It is the ideal generated by $n$-fold products of elements from $I$, and need not be the same as the ideal generated by $n^{th}$ powers of elements from $I$
• In general, the set of products of elements from $I$ and $J$ is not additively closed. An important exception is the situation where either $I$ or $J$ is a principal ideal.
• The product of two ideals is contained in their intersection, and contains the square of their intersection. In symbols:

$(I \cap J)^2 \le IJ \le I \cap J$