Show that $V=Z(x;T)oplus Z(y;T)$ and the $T$-annihilators $mu_{T,x},,mu_{T,y}$ do not share any common divisors implies that $V$ is cyclic

Question

Provided that $V=Z(x;T)oplus Z(y;T)$ where $Z(v;T)$ denotes the cyclic subspace and the corresponding $T$-annihilators $mu_{T,x},,mu_{T,y}$ do not share any common divisors, show that $V$ is itself cyclic.

My approach was to first identify a possible cyclic vector, which was $x+y$ in this case. I then tried to show that every element of $V$ is an element of the cyclic vector space spanned by $T^jx+y, jinmathbb{N}cup {0}$ but the problem seems to be the condition that the $T$-annihilators $mu_{T,x},,mu_{T,y}$ do not share any common divisors.
How do I apply this or how do I continue?

Edit: Definition of the $T$-annihilator as in T-Annihilators and Minimal polynomial :

Definition: $T$-annihilator of a vector $alpha$(denoted as  $p_alpha$) is the unique monic polynomial which generates the ideal
such that $g(T)alpha = 0$ for all $g$ in this ideal.

ancient mathematician · Answer

If the field is $k$, then $pi_x:k[t]to Z(x;T)$ by $p(t)mapsto p(T)x$ is onto with kernel $mu_{T,x}(t)$, so that $Z(x;T)simeq frac{k[t]}{langle mu_{T,x}(t)rangle}$ and similarly for the other factor.
What you want to prove is that, if these minimal polynomials are coprime, then
$$
frac{k[t]}{langle mu_{T,x}(t)rangle}
oplus
frac{k[t]}{langle mu_{T,y}(t)rangle}
simeq
frac{k[t]}{langle mu_{T,x}(t)mu_{T,y}(t)rangle};
$$
this is just the Chinese Remainder Theorem.

Ben Grossmann · Answer

Hint: It suffices to show that $Z(x+y;T)$ contains both $x$ and $y$. To that end, note that the restrictions $mu_{T,x}(T) mid_{Z(y;T)},mu_{T,y}(T) mid_{Z(x;T)}$ are invertible.

Show that $V=Z(x;T)oplus Z(y;T)$ and the $T$-annihilators $mu_{T,x},,mu_{T,y}$ do not share any common divisors implies that $V$ is cyclic

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