Prove that the homomorphism $\phi:R\to S^{-1}R$ is injective if and only if $S$ contains no zero-divisors.
57732
Rings of fractions and homomorphisms
99567
Proving that an inverse ring homomorphism of an ideal is an ideal?
88278
Show $S = f^{-1}(f(S))$ for all subsets $S$ iff $f$ is injective
144928
Find all primes $p$ such that $\phi(x)=x^{13}$ is a homomorphism $\Bbb Z/p\Bbb Z \rightarrow \Bbb Z/p\Bbb Z$.
121395
Let $X$ and $Y$ be non-empty sets and let $f:X \rightarrow Y$. Prove that $ f^{ −1} :P(Y ) → P(X) $ is injective if, and only if, $f$ is surjective.
64025
Ring homomorphisms from $\Bbb Q$ into a ring
149856
Image of ring homomorphism is a subring
97009
Image of ring homomorphism $\phi : \mathbb{Z}[t] \to \mathbb{Q}$?
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
5802
Proving the universal mapping property for polynomial rings
150986
For a Noetherian ring $R$, we have $\text{Ass}_R(S^{-1}M)=\text{Ass}_R(M)\cap \{P:P\cap S=\emptyset\}$
46863
Show that $g$ is injective
30327
$\ker \phi = (x_1-a_1, ..., x_n-a_n)$ for a ring homomorphism $\phi: R[x_1, ..., x_n] \to R$
45179
When does the isomorphism $G\simeq \ker(\phi)\times \operatorname{im}(\phi)$? hold?
108691
Characterize the natural numbers $n$ such that there is a surjective group homomorphism from $S_n$ to $S_{n-1}$.
101064
Showing the triviality of homomorphisms
23735
Divisible module which is not injective