How can I prove that if a domain $R$ is an injective module, then $R$ is a field?
141410
Injective integral domains are fields
58532
If $M$ is an Artinian $R$-module then $R/\mbox{ann}(M)$ is an Artinian ring?
128550
Algebraic field extension and intermediate integral domain
5893
On the commutative property of multiplication (domain of integers, possibly reals)
11202
If $g \circ f$ is injective, so is $g$
148111
Every $R$-module is projective if and only if every $R$-module is injective
16984
Finite extension of domains : preimage of non-zero ideal is non-zero?
11259
Prove $R\times S$ is not an integral domain
101637
Let $R$ be an integral domain. If $x \in R$ is prime, then $x$ is irreducible.
93118
How to show that if two integral domains are isomorphic, then their corresponding field of quotients are isomorphic?
144122
A problem about an $R$-module that is both injective and projective.
93681
Integral closure of $\mathbb{Z}$ in $\mathbb{C}$ is not finitely generated as a $\mathbb{Z}$-module?
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.
551
Why is a finite integral domain always field?
23242
Commutative integral domain with d.c.c. is a field
124672
Let $f:R\rightarrow S$ be a ring homomorphism. Prove or disprove: if $f$ is onto and $R$ is an integral domain, then $S$ is an integral domain
73235
Rings $R/I \cong h(R)/h(I)\,$ for injective ring hom $h$