I have to prove that
1) $\mathbb R[X] /\langle X^2-1\rangle$ is not a field, and
2) $\mathbb R[X,Y]/\langle XY\rangle$ is not a field.
So, I must exhibit an element $r$ from say $\mathbb R[X] /\langle X^2-1\rangle$ that has no multiplicative inverse. I don't know how to do this since I am not so sure what an element of this quotient looks like. One basic doubt that I have is: is $\langle X^2-1\rangle=\langle x^2-1\rangle$? I mean, is the fact that $X$ is in capital letters just a matter of notation?
I suppose that $r=f(x)\langle x^2-1\rangle$, where $f(x) \in \mathbb R[X]$, so $r=f(x)g(x)(x^2-1)$. If $r$ has an inverse, then $rr^{-1}=1_{\mathbb R[X] /\langle X^2-1\rangle}$. Is $\mathbb R[X] /\langle X^2-1\rangle$ a subring of $\mathbb R[X]$? In that case, $1_{\mathbb R[X] /\langle X^2-1\rangle}=1_{\mathbb R[X]}$.
I have the same type of doubts for (2). I would appreciate if someone could explain me how the quotient looks like and to suggest some hint to do the problem.
In both cases they are not integral domains, for example in the first look at $x-1$ and $x+1$.