Questions tagged [injective-module]

Let $A=\mathbb{C}[x_1,\ldots,x_n]$ be a polynomial algebra over the complex numbers. I am interested in injective modules over $A$. Since $A$ is projective over itself, the $\mathbb{C}$-dual module $A^\ast=\mathbb{C}[[x_1,\cdots,x_n]]$ is known to…

Definition: An $R$-module $M$ is called quasi-injective module if for every submodule $N$, any $R$-homomorphism $N\to M$ extends to an endomorphism of $M$. How can I prove that every injective module is quasi-injective ?

Let $R$ be a ring. Suppose that $X$ and $Y$ are left injective $R$-modules and $\theta: X \to Y$ and $\phi:Y \to X$ are $R$-monomorphisms. I want to prove that $X \cong Y$. This is what I have done so far: Since $X \cong \theta(X) \subseteq Y$ and…

I don't think this is the case but I can't find an error in my proof: Let $N \subset M$ be a submodule of an injective module $M$. Suppose we have maps $f:A \to N$ and $h:A \to B$ and we want to prove that there is some map $g:B \to N$ so that $g…