Definition:Power of Ideal of Ring/Definition 2

Definition

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

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

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

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

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

Sources

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