= Solution
{c}
\b[No such property exists.] The <ring isomorphism> in part (b)(ii) identifies <quotient rings> by elements having different <irreducible element> behaviour.
The element $X^3$ is an <irreducible element> of $T$. It is nonzero and a <nonunit>. Every nonzero <nonunit> of $T$ has <polynomial degree> at least $2$: the only possible <polynomials> of smaller <polynomial degree> are <constant polynomials>, and every nonzero <constant polynomial> is a <unit>. If $X^3=uv$ with $u,v\in T$ both <nonunits>, additivity of <polynomial degree> in the <integral domain> $\mathbb R[X]$ would give
$$
3=\deg u+\deg v\geq2+2=4,
$$
which is impossible. Thus one factor must be a <unit>.
In $\mathbb R[Y]$, however, $Y^3=Y\cdot Y^2$ is a product of two <nonunits>, so $Y^3$ is not an <irreducible element>. If a <ring isomorphism invariant> $P$ characterized <irreducible elements> by the indicated <quotient ring>, the <irreducible element> $X^3\in T$ would imply that $T/(X^3)_T$ has $P$. Its isomorphic <quotient ring> $\mathbb R[Y]/(Y^3)$ would then have $P$, incorrectly making $Y^3\in\mathbb R[Y]$ an <irreducible element>.
Consequently,
$$
\boxed{\text{irreducibility cannot be determined from the isomorphism type of }R/(x)}.
$$
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