Prove that it is not a Principal Ideal Ring
3604
If F is a field, then $F[x,y]$ is a Principal Ideal Domain?
155604
Show that $k[x,y]/(xy-1)$ is not isomorphic to a polynomial ring in one variable.
92420
Characteristic of  a Non-unital Integral Ring
117590
Let I and J be ideals of a ring R. Show by example that the set of products {xy | x ∈ I, y ∈ J} need not be an ideal
16602
Proving $M_n(I)\lhd M_n(R)$ and proofs about ideal rings.
151786
Proving that there is no ideal $I$ in $\mathbb{Z}[t]$ such that $\mathbb{Z}[t]/I \cong \mathbb{Q}$
79535
Prove that $R$ is a commutative ring if $x^3=x$
57310
Prove $\mathbb{Z}$ is not a vector space over a field
116883
About ideals in a ring which is a $k$ vector space
149747
Prove that if I is a prime ideal of R then I[x] is a prime ideal of R[x]
22544
Let R be a commutative ring, and let P be a prime ideal of R. Suppose that P has no nontrivial zero divisors in it. Show that R is an integral domain.
104789
How to tell an ideal of integer polynomial ring is principal?
75558
Counter-example for prime ideals in a non commutative ring.
94747
Show that the rings $2\mathbb{Z}$ and $3\mathbb{Z}$ are not isomorphic.
133975
Localising a polynomial ring and non-maximal prime ideal
80961
Prove that a commutative ring $F$ is a field iff $F[x]$ is a Principal Ideal Domain
116000
Prove that prime ideals of a finite ring are maximal