# Questions tagged [ring-homomorphism]

For questions about ring homomorphisms, a function between two rings which respects the structure.

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### Proof verification: Is the quotient ring $\Bbb C[x]/(x^2+1)$ an integral domain? [duplicate]

Is the quotient ring $\Bbb C[x]/(x^2+1)$ an integral domain? My solution goes like this: If possible let us assume that $\Bbb C[x]/(x^2+1)$ an integral domain. This means $(x^2+1)$ is a prime ideal in ...
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### All module homomorphisms from $\mathbb{Z}^n$ to $\mathbb{Z}$

This is a question on something on something more general, but for now I'd like to keep in simple. Consider a module homomorphism $\phi:\mathbb{Z}^n\to\mathbb{Z}$, where $n$ is a positive integer. ...
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### A question about the embedding from $\mathbb{Q}(\sqrt{2},\sqrt{3})$ to an algebraic closure of $\mathbb{Q}$

I am now just beginning my study in field theory and I am trying to find all embeddings from $\mathbb{Q}(\sqrt{2},\sqrt{3})$ to $\bar{\mathbb{Q}}$ (an algebraic closure of $\mathbb{Q}$). Here, an ...
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### What are the module homomorphisms from $\mathbb Q$ to $\mathbb Z/2\mathbb Z$? [closed]

What are the module homomorphisms from $\mathbb Q$ to $\mathbb Z/2\mathbb Z$, looking at $\mathbb Q$ as a $\mathbb Z$-module? The impression I got from the proof in the book is that it is the zero ...
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1 vote
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### homomorphisms from a (semi)local ring to a ring with infinitely many maximal ideals

In continuation of the this question: homomorphism from a (semi)local ring to $\mathbb Z$. I tried to construct (unital) homomorphisms from a (semi)local ring to a ring with infinitely many maximal ...
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1 vote
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### Check finiteness of ring map with SAGE

(I asked this question in a SAGE-specialized forum --see here--, but did not received an answer there sofar. I therefore decided to post the question also here.) Let $R \rightarrow S$ be a ring map. I ...
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### homomorphism from a (semi)local ring to $\mathbb Z$

One can easily construct homomorphisms from $\mathbb Z$ to a (semi)local ring $\mathbb Z/6\mathbb Z$, even a field, $\mathbb Q$. How about the converse? Is there a homomorphism from a (semi)local ...
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1 vote
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### Is the following transformation h from R to S a ring homomorphism?

Do the following ring homomorphisms with identity exist from $R$ to $S$? a) $R=\mathbb{Q}[X]$, $S=\mathbb{Q}$, $f(X^2-2)=0$ b) $R=\mathbb{R}[X]$, $S=\mathbb{C}$, $f(X^2+4)=0$ I know the following ...
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### Let $R$ and $R'$ be two rings and $f:R\to R'$ be a ring homomorphism. Let $I$ be an ideal of $R.$ Is $f(I)$ an ideal of $R'?$ Justify. [duplicate]

Let $R$ and $R'$ be two rings and $f:R\to R'$ be a ring homomorphism. Let $I$ be an ideal of $R.$ Is $f(I)$ an ideal of $R'?$ Justify. I can prove that $f(I)$ is a subgroup of $R'.$ This is because, ...
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### Endomorphisms between direct product/sum of algebraic objects?

My motivation is this Wikipedia article. Suppose $R$ be a ring with unity and $M,N$ be $R$-modules. Take their direct sum/product $P=M \oplus N$. So $P$ is also a $R$-module. Consider the respective ...
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