Difference between revisions of "Characteristic"
(New page: In algebra, the '''characteristic''' of a ring ''R'' is the least natural number ''n'' such that, for all ''r'' in ''R'', ''n'' * ''r'' = 0. If no such natu...) |
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| − | In [[algebra]], the '''characteristic''' of a [[ring (mathematics)|ring]] ''R'' is the least [[natural number]] ''n'' such that, for all ''r'' in ''R'', ''n'' * ''r'' = 0. If no such natural number exists, the ring has characteristic 0. | + | In [[algebra]], the '''characteristic''' of a [[ring (mathematics)|ring]] ''R'' is the least [[natural number]] ''n'' such that, for all ''r'' in ''R'', ''n'' * ''r'' = 0. If no such natural number exists, the ring has characteristic 0. The characteristic of a [[field (mathematics)|field]] must be either 0 or a [[prime number]]. |
| + | ==Examples== | ||
| + | *<math>(\mathbb{Z},+, \cdot)</math> - the set of the integers - has characteristic 0. | ||
| + | *<math>(\mathbb{Z} / 6\mathbb{Z},+, \cdot)</math> has characteristic 6: | ||
| + | :{| | ||
| + | |- | ||
| + | |0 ||= 0||=||0+0+0+0+0+0 | ||
| + | |- | ||
| + | |1+1+1+1+1+1 ||= 0||=|| 1+1+1+1+1+1 | ||
| + | |- | ||
| + | |2+2+2 ||= 0||=|| 2+2+2+2+2+2 | ||
| + | |- | ||
| + | |3+3 ||= 0||=|| 3+3+3+3+3+3 | ||
| + | |- | ||
| + | |4+4+4 ||= 0||=|| 4+4+4+4+4+4 | ||
| + | |- | ||
| + | |5+5+5+5+5+5 ||= 0||=|| 5+5+5+5+5+5 | ||
| + | |} | ||
| + | |||
| + | *If ''R'' is a [[ring with unity]], i.e., there exists a multiplicative identity '''1''', then the characteristic is already given by the smallest ''r'' such that ''r'' * '''1''' = 0. If no such ''r'' exists, the characteristic is defined as zero. | ||
[[Category:Algebra]] | [[Category:Algebra]] | ||
Latest revision as of 05:53, July 13, 2016
In algebra, the characteristic of a ring R is the least natural number n such that, for all r in R, n * r = 0. If no such natural number exists, the ring has characteristic 0. The characteristic of a field must be either 0 or a prime number.
Examples
- - the set of the integers - has characteristic 0.
- has characteristic 6:
0 = 0 = 0+0+0+0+0+0 1+1+1+1+1+1 = 0 = 1+1+1+1+1+1 2+2+2 = 0 = 2+2+2+2+2+2 3+3 = 0 = 3+3+3+3+3+3 4+4+4 = 0 = 4+4+4+4+4+4 5+5+5+5+5+5 = 0 = 5+5+5+5+5+5
- If R is a ring with unity, i.e., there exists a multiplicative identity 1, then the characteristic is already given by the smallest r such that r * 1 = 0. If no such r exists, the characteristic is defined as zero.