253 views

Ascending chain of ideals [duplicate]

Let $R$ be a commutative ring with identity such that every ascending chain of ideals terminate. Let $f:R \to R$ be a surjective homomorphism. Prove that it is an isomorphism.
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$A$ Noetherian and $f:A \rightarrow A$ suryective then $f$ inyective [duplicate]

I think this must have been questioned before, but after searching, I couldn't find it. I thought of considering a set of $\{x_1, ..., x_n\}$ such that $\lt x_1, ... x_n\gt = A$. The hypotesis shows ...
24 views

Hopfian module over commutative ring [duplicate]

I have run into a problem which reads: "Any finitely generated module $M$ over a commutative ring $R$ is Hopfian, i.e., any $R$-epimorphism from $M$ to itself is one-to one." I think the problem ...
2k views

Can you always find a surjective endomorphism of groups such that it is not injective?

If we take the following endomorphism, $\phi:R[t] \to R[t]$ by $\sum_{i = 0}^n a_it^i \mapsto \sum_{i = 0}^{\lfloor n/2 \rfloor} a_{2i} t^i$, it is surjective but not injective. (It just removes odd ...
712 views

Invertible matrices in commutative rings

Let $A$ be a square matrix over a commutative ring $R$. Then $A$ has a left inverse iff it is invertible. Does there exist a elementary proof of this fact? (i.e. without using the determinant!)
499 views

Surjective endomorphism of abelian group is isomorphism

Let $A$ be a finitely generated abelian group and $f:A\rightarrow A$ a surjective homomorphism. How do I prove that $f$ is an isomorphism? And if $f$ were injective instead of surjective would the ...
542 views

This is part of Problem 11.1.6 of Dummit and Foote. The problem reads Let $V$ be a vector space of finite dimension. If $\phi$ is any linear transformation from $V$ to $V$ prove that there is ...
582 views

Surjective endomorphisms of Noetherian modules are isomorphisms.

I'm trying to solve this question: I didn't understand why the hint is true and how to apply it. I really need help, because it's my first question on this subject and my experience on this field is ...
$\phi: A^n \to A^n$ be any injective $A$ linear map,will $\phi$ be surjective?
Let $A$ be a commutative ring with $1$ and $\phi: A^n \to A^n$ be any injective $A$ linear map. Can I say $\phi$ is surjective ? We know about the converse that surjectiveness implies injectivitness,...