Radical ideal computation (Macaulay2) 
Is there a way to find the radical ideal of $I_i=(a^n-u^{n-i+1}v^{n-i},
b^n-u^{i-1}v^i, uv-ab)$ for $1\leq i \leq n$ in $\mathbb{C}[u,v,a,b]?$

This is the generalization of my question here where I wanted to use Macaulay2 software to compute the radical ideal for $n=3$ and $i=2$ of the above ideal. Unfortunately, I don't know how to use the software in the general case, and I don't know if it works or not. At least, using Macaulay2 for some special cases I can guess that $\sqrt{I_i}=(a^i-u^{n-i}b^{i-1}, b^{n-i+1}-v^{i-1}a^{n-i}, uv-ab)$ but there is problem in my further computation, so I thought maybe what I guessed is wrong! 
I would appreciate any help on that. 
Motivation:
This is indeed related to the Derived McKay correspondence where I'm studying the image of torus-invariant, zero-dimensional sheaves of the minimal resolution $Y$ of $\mathbb{C}^2/\mathbb{Z}/n$ under the Fourier-Mukai transform from the (bounded) derived category of coherent sheaves on $Y$ to the (bounded) derived category of coherent sheaves on $\mathfrak{X}=[\mathbb{C}^2/\mathbb{Z}/n],$ the stacky resolution of $\mathbb{C}^2/\mathbb{Z}/n.$  
 A: Is this roughly what you want to do?

Script: ~/tmp/ideals.m2
R = QQ[u,v,a,b];

n = 10;

myIdeal = (i,n) -> (
    p1 = a^n - (u^(n-i+1))*(v^(n-i));
    p2 = b^n - (u^(i-1))*(v^i);
    p3 = u*v - a*b;
    ideal(p1,p2,p3)
    )

myRad = (I) -> (
    radical I
)

for j from 1 to n do (
    J = myIdeal(j,n);
    rJ = myRad(J);
    print rJ
)


Output: (for $1 \leq i \leq 10$)
ii60 : load "~/tmp/ideals.m2"
                 10           9
ideal (- u*v + a*b, b   - v, - u*b  + a)
                     9             8    2
ideal (- u*v + a*b, b  - v*a, - u*b  + a )
                     8      2       7    3
ideal (- u*v + a*b, b  - v*a , - u*b  + a )
                     7      3       6    4
ideal (- u*v + a*b, b  - v*a , - u*b  + a )
                     6      4       5    5
ideal (- u*v + a*b, b  - v*a , - u*b  + a )
                     6      4       5    5
ideal (- u*v + a*b, a  - u*b , - v*a  + b )
                     7      3       6    4
ideal (- u*v + a*b, a  - u*b , - v*a  + b )
                     8      2       7    3
ideal (- u*v + a*b, a  - u*b , - v*a  + b )
                     9             8    2
ideal (- u*v + a*b, a  - u*b, - v*a  + b )
                     10           9
ideal (- u*v + a*b, a   - u, - v*a  + b)


Example: ($i=2$, $n=3$)
ii61 : J2 = myIdeal(2,3)

                 2     3       2    3
oo61 = ideal (- u v + a , - u*v  + b , u*v - a*b)

oo61 : Ideal of R

ii62 : myRad(J2)

               2               2
oo62 = ideal (a  - u*b, v*a - b , u*v - a*b)

oo62 : Ideal of R

A: To compute the radical of a binomial ideal, in some cases it can be advantageous (that is, faster) to use the Binomials package in Macaulay2. Here is the documentation.  In Charles Boyd's answer you can just replace the call to radical by binomialRadical.
