Product of ideals

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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