# Questions tagged [ring-isomorphism]

This tag is for questions regarding to Ring Isomorphisms, a ring homomorphism having a $2$-sided inverse that is also a ring homomorphism. Isomorphic rings have all their ring-theoretic properties identical. One such ring can be regarded as "the same" as the other.

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### Is this set a ring?

I am studying Ring Theory for the first time in my life- so the following question may be a very silly one. While trying to solve an (unrelated) exercise, this question clicked in me, and it has been ...
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### Every isomorphism of commutative ring with 1 with product of two rings is defined by idempotent element $x$ and element $1-x$. [duplicate]

I proved that $f: R \rightarrow P_x \times P_{x\prime}$ where $x$ is impodent element in $R$, $x\prime=(1-x)$ (also impodent) and $P_x=\{rx:r \in R\}$. But I have problem with proving that every ...
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### How to come up with this isomorphism?

A question from Herstein's Abstract Algebra book goes- Let $(R,+,\cdot)$ be a ring with unit element. Using its elements we define a ring $(\tilde R,\oplus,\odot)$ by defining $a\oplus b = a + b + 1$ ...
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### Given a proper field extension $L/K$, can we have $L\cong K$? [duplicate]

Given a proper field extension $L/K$ (that is, $K$ can be considered as a proper sub-field of $L$). Can it still happen that $K\cong L$ via a field-isomorphism? I assume No, but I am utterly ...
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### Prove: $F[t]/H[t]$ is isomorphic to $(F/H)[t]$

Let $F$ a commutative and unitary ring, and also $F$ is an integral domain. Let $H$ be a proper ideal of $F$. I want to prove: $F[t]/H[t]$ is isomorphic to $(F/H)[t]$ So my idea is to use the ...
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### Proof for $P(R) \leq I-rad(R_R)$

The (right) $\textbf{isoradical}$ $I-rad(R_R)$ of a ring $R$ is the intersection of the annihilators of all isosimple right $R-$modules where an $\textbf{isosimple}$ module is defined as a non-zero ...
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### Completely virtually semisimple modules are direct sums of isosimple modules.

An $\textbf{isosimple}$ module is defined as a non-zero module whose all non­zero submodules are isomorphic to it. An $R$-module $M$ is called $\textbf{virtually semisimple}$ if every submodule of $M$ ...
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### $\sigma:Z\to K$ is an integral domain. Why is this an explanation for the fact that $\sigma(Z)\cong Z$ or $F_p$?

Someone mentioned Serre's A Course in Arithmetic in a book-recommendation question. I was curious and so I downloaded a copy and read it, but I was already tripped by the opening paragraph in the ...
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### Prove that $\mathbb Z[i]/(1+3i)\simeq \mathbb Z_{10}$ [duplicate]

I am confused how to do the proof that $\mathbb Z[i]/(1+3i)\simeq \mathbb Z/10\mathbb Z$. It is clear to me intuitively. $1+3i=0$ in the quotient ring, so $3i=-1\implies 9i^2=1\implies 10=0$ in ...
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### How to prove this isomorphism?

I know about the correspondence between ideals in $A$ that contains $\mathfrak a$ and ideals in the quotient ring $A/\mathfrak a$, but never see the isomorphism above, anyone knows how to prove it?
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### If $f :R \to S$ is an isomorphism from ring $R$ to ring $S$, both with identity, does $f(1_R) = 1_S$?

My attempted proof: Let $a \in R$, then $f(a) = f(a \cdot 1_R)$ = $f(a) \cdot f(1_R)$, and for $f(a) \ne 0$, it gets canceled, so $1_S = f(1_R)$ Is this correct? Thanks.
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### Proof of Dedekind-Kummer theorem

I will write the statement and then ask my query about part of the proof. Statement Let $p$ be a rational prime. Let $K=\mathbb{Q}(\theta )$ be a number field where $\theta$ is an algebraic integer. ...
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### isomorphism $\psi$ between quotient rings

Show that there is an isomorphism $\psi$ between the rings $R_1 := \mathbb{Q}[r,s]/(r+s-1, r-r^2, s-s^2)$ and $R_2 := \mathbb{Q}[y]/(y^2-1)$ so that $\psi([r]) = [(1+y)/2]$ and $\psi(s)=[(1-y)/2].$ I ...
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### Show that $\text{Frac}(\mathbb{Z}(\sqrt{2}))$ is isomorphic to $\mathbb{Q}(\sqrt{2})$

Show that $\text{Frac}(\mathbb{Z}(\sqrt{2}))$ is isomorphic to $\mathbb{Q}(\sqrt{2})$. Here $\text{Frac}(\mathbb{Z}(\sqrt{2}))$ is the fiel of fractions of $\mathbb{Z}(\sqrt{2})$. I already proved ...
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### $\mathbb{R}[x,y]/(x^3,y) \cong \mathbb{R}[x,y]/(x^2,y^2)$

I have to prove this ring isomorphism as an exercise. What I can say is that $\mathbb{R}[x,y]/(x^3,y)$ is made of the polynomials of the form $a_0 + a_1 \bar x + a_2 \bar x^2$ (with $\bar x^3 = 0$) ...
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### Show isomorphism between localization of rings

I have to show that this two rings are isomorphic $$(\mathbb{Q}[X,Y]/\langle XY-1\rangle)_{\langle X-1,Y-1 \rangle} \simeq \mathbb{Q}[X]_{\langle X-1\rangle}$$ I tried using the fact that  \mathbb{Q}...
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### Subring of $\mathbb{Z} \times \mathbb{Z}$

Let $R = \mathbb{Z} \times \mathbb{Z}$ and let $S$ be a subring of $R$ isomorphic to $R$. Show that $S = R$. I know that as $S$ is subring, $S \subseteq R$. But after this I only get that $A$ is ...