Colon of Two ideals in a ring

Question

Is there any example of a commutative ring $R$ with unity and its proper ideal $I$ where $(I^{k+1} : I) = I^k $ does not hold for some value of k ?

Answer 1

There might be a simpler answer, but here's one I came up with (assuming you wanted an example where $k\neq 0$).

Let $\mathbb{F}$ be some field, and let $R= \mathbb{F}[x,y]$. Now let $I=(x^4, x^3y, xy^3, y^4)$, that is, all degree 4 monomials except for $x^2y^2$.

Then $I^2 = (x^8, x^7y, x^6y^2, x^5y^3, x^4y^4,x^3y^5,x^2y^6, xy^7,y^8)$, that is, all degree $8$ monomials.

Then notice that $x^2y^2$ is in $(I^2:I)$, because multiplying any element of $I$ by a degree 4 monomial will surely get you a degree 8 monomial. But $x^2y^2$ was not in $I$ to begin with.

So $(I^2:I) \supsetneq I$.