If the intersection of all non-zero ideals of an integral domain R is non-zero, show that R is a field.
134773
A domain is a field if it has a common multiple $\!\neq\! 0$ of all elements $\!\neq\! 0$
7828
Proof that an integral domain that is a finite-dimensional $F$-vector space is in fact a field
74370
$P$ is a prime ideal of $R$ iff $R/P$ is an integral domain. : $P≠R$
62951
Is there a more direct way of proving that this ring is an integral domain?
96787
Why is the arbitrary sum, but not the arbitrary intersection, of ideals an ideal?
19838
Prove that in general the set of the right zero divisors is different from the set of left zero divisors in a ring R
57669
A proof that $\mathbb{Z}_{p}$ is an integral domain if and only if $p$ is prime.
122415
How to show every field is a Euclidean Domain.
62134
Show that if $ R $ is an integral domain then a polynomial in $ R[X] $ of degree $ d $ can have at most $ d $ roots
90314
Maximal ideals in $R[x]$
94819
Prove that $\frac{\mathbb{C}[x,y]}{\langle x^2-y^2-1\rangle}$ is an integral domain such that all the nonzero prime ideals are maximal.
82576
Intersection of two subfields of the Rational Function Field in characteristic $0$
95146
Polynomial ring $R[X] $ has zero Jacobson radical, for $R$ a domain
50908
Showing a Ring and an Ideal are equal
101637
Let $R$ be an integral domain. If $x \in R$ is prime, then $x$ is irreducible.
81296
If A and B are ideals of a ring, show that A + B = $\{a+b|a \in A, b \in B\}$ is an ideal
96132
$R$ integral domain : $u\in R^*, a \text{ is prime} \iff au \text{ is prime}$