# Questions tagged [ring-homomorphism]

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

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### Proof Verification: Hom(Z[x],S)=S (as rings)

How to prove Hom(Z[x],S)=S (as rings), where S is any ring? My attempt: took an element b in S, defined a map , b: Z[x]-> S which maps f(x) to f(b). Clearly b is a ring homomorphism, hence we ...
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### Isomorphic $K[x]$-modules have equal characteristic polynomials

Let $K$ be a field, $V,V'$ be finite dimensional $K$-vector spaces and $A\in \text{End}_K(V),A'\in \text{End}_K(V').$ Regard $V$ (respectively $V'$) as $K[x]$-modules with respect to $A$(resp. $A'$.) ...
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### Finite-Type Algebra Property Under Change of Base

I was wondering how to prove the following assertion. Any help would be appreciated! Suppose we have to ring homomorphisms: $f: A \rightarrow B$ and $g: B \rightarrow C$. Then $C$ being finite type ...
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### Non-trivial homomorphism from $\mathbb{Z}[x]/\langle x^n-1 \rangle$ to $\mathbb{Z}_m[x]/\langle x^n+1 \rangle$?

Is there any non-trivial homomorphism from $R_1=\mathbb{Z}[x]/\langle x^n-1 \rangle$ to $R_2=\mathbb{Z}_m[x]/\langle x^n+1 \rangle$, for all $m>1$ (no necessarily prime)? (Note that the elements in ...
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### Primary decomposition of an ideal and its extension

I'm trying to solve a problem in Sharp's Steps in Commutative Algebra, to be precise Exercise 4.22 which states the following: Let $f:R \rightarrow S$ be a surjective homomorphism of commutative ...
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### Subjectivity of evaluation homomorphism

I am trying to prove that the evaluation homomorphism: $ev_z:\mathbb{R}[X]\to \mathbb{C}, f\mapsto f(z)$ where $z=a+bi$ is surjective. To start with, I don't really understand how the map is ...
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### $\varphi(u)$ is invertible in $R'$ iff no invertible element of $R$ belongs to $\ker \varphi$.

Assume the surjective ring homomorphism $\varphi: R \longrightarrow R'$, where $R$ is a ring with unity. Also let $u$ be an invertible element of $R$. Show that $\varphi(u)$ is invertible in $R'$ ...
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### A module over PID [closed]

Let R be a PID and M be a module over R ⋀ cyclic. Then prove that ∀N⊂M; a submodule of M, N is cyclic. [My idea] ∃m∈M s.t. M=Rm (∵M is cyclic) p: R(= a module over R) → M, p(a) = am; module-hom ⋀ ...
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### Is this proof on ideal maximality correct?

The exercise is: Given that $\phi : R \to S$ is an onto ring homomorphism, let $B$ be a maximal ideal of $S$. Prove that $A = \phi^{-1} (B)$ is a maximal ideal of $R$. Ok, I previously ...
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### If $\varphi:R\to S$ be a surjective ring homomorphism then $\varphi(R^*)$is not always $S^*$ [closed]

I don't know any example. Whatever examples I know the statement holds true
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### Direct Product of a Clean Rings

Commutative rings whose elements are a sum of an unit and idempotent by Anderson & Camillo (2002) Definition 1. A commutative ring $R$ is a clean ring if every element $x\in R$ can be written in ...
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### Existence of Ring homomorphism from Formal Power series ring to a ring.

Let $R$ be a ring, and $S$ be a subring of $R$. Denote $S[[x]]$ for a ring of formal power series with coefficients in $S$. Let $\alpha \in R$ be a unit, such that $\alpha \notin S$. Can there exist a ...
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This question originates from Pinter's Abstract Algebra, Chapter 24, G7. Let $h:\mathbb{Z}\rightarrow\mathbb{Z}_n$ be a homomorphism with kernel $K$. Define $\bar{h}: \mathbb{Z}[x]\rightarrow\... 0answers 6 views ### Can we define a mapping from the set of graphs to the set of adjacency matrices? Graph union: the union of two graphs$G_1=(V_1, E_1)$and$G_2=(V_2, E_2)$is defined and denoted by$G_1\cup G_2=(V_1\cup V_2, E_1\cup E_2)$. Graph intersection: the intersection of two graphs$G_1=(...
Let $x$ be a fixed real number and let $\mathbb{Q}$ be the field of rational numbers. It is known that $\mathbb{R}$ is an extension field of $\mathbb{Q}$ and thus there exists some homomorphisms. Can ...