TITLE: Find the isomorphic ring and its order with $\mathbb{Z_{15}}[x] / \langle 3x^2 + 5x \rangle $
QUESTION [1 upvotes]: 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

REPLY [0 votes]: 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 $\Rightarrow E/3  \cong \Bbb Z_3[x]/5x,\,$ $\,E/5  \cong \Bbb Z_5[x]/3x^2$
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