Characteristic (algebra) – Wikipedia, the free encyclopedia

In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring\’s multiplicative identity element (1) in a sum to get the additive identity element (0); the ring is said to have characteristic zero if this sum never reaches the additive identity.

That is, char(R) is the smallest positive number n such that

underbrace{1+cdots+1}_{n text{ summands}} = 0

if such a number n exists, and 0 otherwise.

via Characteristic (algebra) – Wikipedia, the free encyclopedia.

Ideal (ring theory) – Wikipedia, the free encyclopedia

For an arbitrary ring (R,+,cdot), let (R,+) be its additive group. A subset I is called a two-sided ideal (or simply an ideal) of R if it is an additive subgroup of R that \”absorbs multiplication by elements of R\”. Formally we mean that I is an ideal if it satisfies the following conditions:

(I,+) is a subgroup of (R,+)

forall x in I, forall r in R :quad x cdot r in I

forall x in I, forall r in R : quad r cdot x in I.

Equivalently, an ideal of R is a sub-R-bimodule of R.

via Ideal (ring theory) – Wikipedia, the free encyclopedia.