Category:Definitions/Powers of Ideals of Ring
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This category contains definitions related to Powers of Ideals of Ring.
Related results can be found in Category:Powers of Ideals of Ring.
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 $R$.
Pages in category "Definitions/Powers of Ideals of Ring"
The following 3 pages are in this category, out of 3 total.