Proving rI is a Minimal Right Ideal in R

[jem05]

Homework Statement

R is a ring, r in R.
I is a minimal right ideal of R
rI =/= 0
prove rI is a minimal right ideal.

Homework Equations

rI minimal

The Attempt at a Solution

i proved rI right ideal, but I am having trouble with minimal, the thing is Ir is not a left nor a right ideal.
i tried supposing that there is a K right ideal not null and nor = rI but between them, and tried to consruct a right ideal that is neither null nor = I but also between them, to get a contradiction, but i failed.
any hint is appreciated.
thank you

 

[arkajad]

Suppose K is a non-zero right ideal contained in rI. What about defining

$$I_0=\{i\in I:\, ri\in K\}$$

and proving that it is a right ideal using the fact that I, K are right ideals?