Find the isomorphic ring and its order with $\mathbb{Z_{15}}[x] / \langle 3x^2 + 5x \rangle $

Question

Here is my attempt

this question realated with Product of the ideal and normal groups(Is this solution right?)

$sol)$ the solution is similar in the above link.

Let $f(x) = 3x^2 + 5x$ and $[f]_n$ be $f(x)(mod n)$

Then $\mathbb{Z_{15}}[x] / \langle 3x^2 + 5x \rangle $ = $\mathbb{Z_{15}}[x] / \langle [f]_{15} \rangle $ for $\langle [f]_{15}(= 3x^2 + 5x) \rangle \lhd\mathbb{Z_{15}}[x]$

Here , $\mathbb{Z_{15}}[x] \simeq \mathbb{Z_{3}}[x] \times \mathbb{Z_{5}}[x]$

Plus By C.R.T, $\langle [f]_{15} \rangle \simeq \langle ([f]_3, [f]_5) \rangle \lhd \mathbb{Z_{3}}[x] \times \mathbb{Z_{5}}[x] $

By the way, Since $\mathbb{Z_{15}}[x] (\simeq \mathbb{Z_{3}}[x] \times \mathbb{Z_{5}}[x])$ is a ring with unity, $\langle ([f]_3, [f]_5) \rangle = \langle [f]_3 \rangle \times \langle [f]_5) \rangle$

I.e. $\langle [f]_{15} \rangle \simeq \langle ([f]_3, [f]_5) \rangle = \langle [f]_3 \rangle \times \langle [f]_5) \rangle$

So, ($\mathbb{Z_{3}}[x] \times \mathbb{Z_{5}}[x]$) / $\langle ([f]_3, [f]_5) \rangle $ $\simeq$ $(\mathbb{Z_{3}}[x] / \langle [f]_3 \rangle) \times $$(\mathbb{Z_{5}}[x] / \langle [f]_5 \rangle $)

But I'm not able to say the point "$\color{red}\simeq$" in the above link that first statement is not always true.

$\mathbb{Z_{15}}[x] / \langle [f]_{15} \rangle $ $\color{red}\simeq$ ($\mathbb{Z_{3}}[x]$ $\times$ $\mathbb{Z_{5}}[x]$)/$\langle [f]_3, [f]_5 \rangle$ $\simeq$ $(\mathbb{Z_{3}}[x] / \langle [f]_3 \rangle) \times $$(\mathbb{Z_{5}}[x] / \langle [f]_5 \rangle $)

What should I next? or Are there any different ways without my method?

One thing sure is the order of quotient ring is 75

please help

Answer 1

Hint: $ $ in $\,R = \Bbb Z/15\!:\ (3)+(5)=(1)\,\Rightarrow\, (3)\cap (5) = (3)(5) = (0)$ $\smash{\overset{\small\rm CRT}\Rightarrow}\, R^{\phantom{|^|}}\!\!\! \cong R/3\times R/5$

The above ideal equalities extend to $\,E = R[x]/(3x^3+5x)\,$ thus also $\smash{\overset{\small\rm CRT}\Rightarrow}\, E^{\phantom{|^|}}\!\!\! \cong E/3\times E/5$

Ring isomorphism theorems $\Rightarrow E/3 \cong \Bbb Z_3[x]/5x,\,$ $\,E/5 \cong \Bbb Z_5[x]/3x^2$