Quotient ring of polynomials

Question

Is it true $Q[x]/\langle(x-1) \rangle \subset $ $Q[x]/\langle(x-1)(x-2) \rangle$ ?

What do you think about that?

Plus Is $Q[x]/\langle(x-1) \rangle$ a subring or ideal of $Q[x]/\langle(x-1)(x-2) \rangle$?

I thought like the below but I'm not sure whether right or not.

Here is a My thought

As a point of view of the set, the two set are disjoint hence neither subset nor subring(or ideal).

Answer 1

In general if $I,J$ are ideals of ring $R$ with $I\subseteq J$ then the function $\nu:R/I\to R/J$ prescribed by $r+I\mapsto r+J$ is a well defined surjective ringhomomorphism.

This reveals that $R/J$ is isomorphic with a quotient of $R/I$.

Observe that the underlying sets of these rings are distinct if $I\neq J$.

So it is out of the question then that one is a subring of the other.

Applying that on:

• $R=\mathbb Q[x]$,
• $I=\langle(x-1)(x-2)\rangle$
• $J=\langle(x-1)\rangle$

we find that $\mathbb Q[x]/\langle(x-1)\rangle$ can be looked at as (isomorphic with) a quotient of $\mathbb Q[x]/\langle(x-1)(x-2)\rangle$.

Answer 2

The third isomorphism theorem shows that for a commutative ring $R$ and ideals $I$ and $J$ of $R$ with $I\subseteq J$, $R/J$ is isomorphic to $(R/I)/(J/I)$. So as said above, $R/J$ is isomorphic to a quotient ring of $R/I$.