Yes, you can use the same definition for a non-commutative associative ring R. Note, however, that the resulting Nagata extension will typically be a non-associative ring!

For arbitrary (a,m), (b,n), (c,p) the equation ((a,m)*(b,n))*(c,p)=(a,m)*((b,n)*(c,p)) is equvalent to

n\sigma(a)c + mbc + p\sigma(ab) = nc\sigma(a) + mbc + p\sigma(ba).

If R is commutative, then the above equation is obviously satisfied, but if R is non-commutative then we can not immediately tell.

I'm agree with Johan Öinert. I thint that the Nagata extension is possible defined for the noncommutative rings too.

Thank you all for your collaboration