Difference between revisions of "Ideal (mathematics)"
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#The ideal is closed under addition | #The ideal is closed under addition | ||
#The product of an element of the ideal and an element of the initial ring is an element of the ideal | #The product of an element of the ideal and an element of the initial ring is an element of the ideal | ||
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| + | == Examples == | ||
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| + | An example is the set of even integers, as a subset of the ring of integers. This is an ideal because: | ||
| + | # 0 is even. | ||
| + | # The sum of two even integers is even. | ||
| + | # The product of an even integer with and any other integer is even. | ||
[[Category:Mathematics]] | [[Category:Mathematics]] | ||
Revision as of 20:39, October 4, 2009
In ring theory, an ideal is defined as a subset of a ring with the following properties:
- Zero is an element of the ideal
- The ideal is closed under addition
- The product of an element of the ideal and an element of the initial ring is an element of the ideal
Examples
An example is the set of even integers, as a subset of the ring of integers. This is an ideal because:
- 0 is even.
- The sum of two even integers is even.
- The product of an even integer with and any other integer is even.