Showing that $R[x_1,...,x_n]$ is the only $R$-algebra up to isomorphism that satisfies the universal property of the polynomial ring.
5802
Proving the universal mapping property for polynomial rings
119297
Finding invertible polynomials in polynomial ring $\mathbb{Z}_{n}[x]$
53244
Prove that $p \in \mathbb{R}[x]$ can be represented as a sum of squares of polynomials from $\mathbb{R}[x]$
30327
$\ker \phi = (x_1-a_1, ..., x_n-a_n)$ for a ring homomorphism $\phi: R[x_1, ..., x_n] \to R$
111248
The Uniqueness of a Coset of $R[x]/\langle f\rangle$ where $f$ is a Polynomial of Degree $d$ in $R[x]$
121244
Homomorphisms and automorphisms on polynomial rings
78950
Does this map extend to a unique automorphism of the polynomial ring of an integral domain?
126295
What is an $R$-algebra?
67706
If $R$ is a Noetherian ring then $R[[x]]$ is also Noetherian
21578
Showing the minimal polynomial for an element in an extension field is the same as the minimal polynomial of a linear transformation.
146715
A square matrix has the same minimal polynomial over its base field as it has over an extension field
28658
$R$ with an upper bound for degrees of irreducibles in $R[x]$
145221
Ring isomorphism (polynomials in one variable)
98897
Proving that the set of $\nu$ measurable sets $M\subseteq P(X)$ is an algebra
46843
A ring without the Invariant Basis Number property
104782
Polynomial irreducible - maximal ideal
49854
Ideals in polynomial rings generated by linear polynomials