Let $\phi :F \to R $ be a ring homorphism. Prove either $\forall x \in F ,\phi (x)=0 $ or $\phi$ injective
75315
How to prove that homomorphism from field to ring is injective or zero?
39576
Quotient ring being isomorphic to the initial ring
158282
Finite order of an element $f(g)$ of a homomorphism $f$
111239
Are monomorphisms of rings injective?
38691
Injectivity of Homomorphism in Localization
11202
If $g \circ f$ is injective, so is $g$
60760
Ring homomorphism and characteristic
44396
Noetherian Ring under a Homomorphism / homomorphic function / map
51483
Prove: If a linear map, $φ: U→V$ is bijective, then $φ(x)=0$ has only the solution $x=0$.
148813
Prove: $f: \mathbb{R} \rightarrow \mathbb{R}$ st for every $x \in \mathbb{R}$ there exists $n$ st $f^{(n)}(x) = 0$, f is a polynomial.
119702
Showing a bijection between the set of ringhomomorphisms $\mathbb{Z}[i] \to R$ and the set of $x \in R$ for which $x^2+1=0$
125582
Let $\phi:G_1\to G_2$ be a group homomorphism. Show $\phi(g^{-1})=(\phi(g))^{-1}$.
16092
How to prove that the composition of two surjective functions is surjective
73235
Rings $R/I \cong h(R)/h(I)\,$ for injective ring hom $h$
45179
When does the isomorphism $G\simeq \ker(\phi)\times \operatorname{im}(\phi)$? hold?
31216
Is the image of $\Phi_n(x) \in \mathbb{Z}[x]$ in $\mathbb{F}_q[x]$ still a cyclotomic polynomial?
30714
A question on ring homomorphisms and maximal ideals.