Ideal (ring theory)

If R is a ring and I is a subset which is a subgroup of R under

addition, and which also has the property that if a is in R

and x is in I then ax is in I, then I is called a left ideal

of R. Right ideals are defined similarly and if I is both it

is called a two-sided ideal. Of course if R is commutative

left, right and two-sided ideals are the same concept.

One reason that ideals are important is that if f:R-->S

is a ring homomorphism, that is, a map which satisfies

f(a+b)=f(a)+f(b), f(ab)=f(a)f(b) and f(1)=1, then the

kernel I of f, that is, the set of x in R such that

f(x)=0 is a two-sided ideal, and conversely given a

two-sided ideal I of R, the quotient group R/I has a

natural ring structure.