Characteristic

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

• $(\mathbb {Z} ,+,\cdot )$ - the set of the integers - has characteristic 0.
• $(\mathbb {Z} /6\mathbb {Z} ,+,\cdot )$ 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.