$\mathbb{R}[x]$ module generated by the following generators and relations

I am trying to compute $$\langle a,b,c\mid xa=b,b=xc\rangle$$ as a module generated over $\mathbb{R}[x]$ (with relations).

Is the following result and working correct?

\begin{align} \langle a,b,c\mid xa=b,b=xc\rangle&=\langle a,c\mid xa=xc\rangle\\ &=\langle a,a-c\mid x(a-c)=0\rangle\\ &=\mathbb{R}[x]\oplus\mathbb{R}[x]/(x) \end{align}

where $(x)$ denotes the principal ideal generated by $x$.

Thanks.

• Your calculations are alright. – MooS Sep 28 '17 at 11:42
• @Moos: You are absolutely right, serious brainfart on my end! I don't know what I was thinking. – Mathematician 42 Sep 28 '17 at 11:43