Definition:Power of Ideal of Ring

Definition

Let $\struct {R, +, \circ}$ be a ring.

Let $I$ be an ideal of $R$.

Let $n \in \Z$ be an integer such that $n \ge 1$.

Definition 1

The $n$th power of $I$ is the set of all finite sums of the products of $n$ elements of $I$:

$\ds I^n = \set {\sum_{i \mathop = 1}^r a_{i, 1} \cdots a_{i, n} : r \in \N, a_{i, j} \in I}$

where $a_{i, j}$ denotes a doubly indexed term of the family $\family a_{r, n}$.

Definition 2

The $n$th power of $I$ is the ideal generated by products of $n$ elements of $I$.

That is:

$\ds I^n = \gen {\prod_{i \mathop = 1}^n a_i : a_i \in I}$

where the $\ds \prod_{i \mathop = 1}^n$ notation denotes a continued product on the ring product of $R$.

Also see

• Equivalence of Definitions of Power of Ideal of Ring
• Definition:Product of Ideals of Ring

• Results about powers of ideals of ring can be found here.

Sources

• 2003: David S. Dummit and Richard M. Foote: Abstract Algebra (3rd ed.): $\S 7.3$