It is not true that ideal product is commutative in non-commutative (associative, unital) rings. An easy example is given by triangular rings (see for instance pp 17-22, especially p. 18 of Lam's First Course in Non-commutative Rings).
You can easily construct a finite counterexample with 8 elements: Let a be the ring of upper-triangular matrices; let i be the (two-sided) ideal of a consisting of strictly upper triangular matrices, and let j be the (two-sided) ideal consisting of matrices whose second row is zero. Then ij = 0, but ji = i. This works over any field, in particular the field with 2 elements.
You can verify this in GAP:
gap> a:=AlgebraWithOne(GF(2),[[[0,1],[0,0]],[[1,0],[0,0]]]*One(GF(2)));
gap> i:=Ideal(a,[[[0,1],[0,0]]]*One(GF(2)));
, (1 generators)>
gap> j:=Ideal(a,[[[1,0],[0,0]],[[0,1],[0,0]]]*One(GF(2)));;
gap> ij:=Ideal(a,Concatenation(List(i,x ->List(j,y->x*y))));;
gap> ji:=Ideal(a,Concatenation(List(i,x ->List(j,y->y*x))));;
gap> Dimension(ij);Dimension(ji);
0
1
gap> ji = i;
true