Right Ideal

In a noncommutative ring , a right ideal is a subset which is an additive subgroup of and such that for all and all ,

(1)

For all , the set

(2)

is a right ideal of , called the right ideal generated by .

In the ring of matrices with entries in , the subset

(3)

is a right ideal. This is evidently an additive subgroup, and the multiplication property can be easily checked,

(4)

It is not a left ideal, since

(5)

but

(6)

In this example, is a one-sided ideal which is not two-sided.