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.