If $A\subset B$ are integral domains with the same field of fractions and $B$ is faithfully flat over $A$, then $A=B$.
53531
Birational and faithfully flat $\implies$ isomorphism
58285
Is it true that if $X$ is connected, then for every nonempty proper subset $A$ of $X$, we have $\mathbf{Bd} \ne \emptyset$
10909
$|A \cup B| = \mathfrak{c}$, then $A$ or $B$ has cardinality $\mathfrak{c}$.
116453
Is any quotient of a Euclidean domain by a prime ideal a Euclidean domain?
70193
Show that the relation defined on an integral domain by "$x\sim y$" ("$x$ is associated with $y$") is a relation of equivalence.
77735
When $\Bbb Z_n$ is a domain. Counterexample to $ab \equiv 0 \Rightarrow a\equiv 0$ or $b\equiv 0\pmod n$
142699
Is $R/N(R)$ a faithfully flat $R$-module?
8317
Understanding a proof of "If $R$ is an integral domain, then $R[x]$ is an integral domain as well".
551
Why is a finite integral domain always field?
95408
Give an example to show that a factor ring of a ring with divisors of 0 may be an integral domain
18887
Euclidean domain in which the quotient and remainder are always unique
112345
In an Integral Domain is it true that $\gcd(ac,ab) = a\gcd(c,b)$?
83009
Showing that $R(x)$ is a proper subset of $R((x))$ if $R$ is a field
50908
Showing a Ring and an Ideal are equal
131983
Prove that $A \subset B$ if and only if $A \setminus B = \emptyset$
85206
Verifying proof :an Ideal $P$ is prime Ideal if $R/P$ is an integral domain.
71479
Is there some elementary proof of invariance of domain?