Quotient of two Free Abelian Groups (Checking Computation)

Question

Let $A=\langle a-b,c\rangle$ be the free abelian group generated by $a-b$ and $c$.

Let $B=\langle 2a-2b+c,2c\rangle$ denote the free abelian group generated by $2a-2b+c$ and $2c$.

I would like to compute the quotient $A/B$. I got $\mathbb{Z}_4\oplus\mathbb{Z}_2$ as such:

Quotienting by $B$ is implies $2c=0$ and $2a-2b+c=0$. This implies $c=2b-2a$, so $4(b-a)=0$.

So $c$ generates $\mathbb{Z}_2$ and $a-b$ generates a $\mathbb{Z}_4$. Is the above correct?

Thanks.