Show that $\mathbb{C}$ and $\mathbb{R[x]}/\langle x^2+1\rangle$ as $\mathbb{R}$-Algebras are isomorphic.  This is the first time that we are dealing with polynomial rings and algebras so i am a bit confused. The first problem is that I can't wrap my head around what elements of $\mathbb{R[x]}/\langle x^2+1\rangle$ are. I kinda understand theoretically but can't practically. The second problem is to find the isomorphism. Now if $\langle x^2+1\rangle$ is an Ideal of $\mathbb{R[x]}$ then I guess there is a theorem which implies that they are isomorphic. Si it remains only to show that it is an ideal? Thanks in advance
123353
Show that $\Bbb R[x] / \langle x^2 + 1 \rangle$ is isomorphic to $\Bbb C$.      Let $\phi$ be the homomorphism from $\Bbb R[x]$ onto $\Bbb C$ given by   $f(x) \rightarrow f(i)$ (that is, evaluate a polynomial in $\Bbb R[x]$   at $i$).  Then $x^2 + 1 \in \operatorname{Ker} \phi$ and is clearly a   polynomial of minimum degree in $\operatorname{Ker} \phi$.  Thus,   $\operatorname{Ker} \phi = \langle x^2 + 1 \rangle$ and $\Bbb R[x] /\langle x^2 + 1 \rangle$ is isomorphic to $\Bbb C$.   My question is:  How is $x^2 + 1$ a polynomial of minimum degree in $\operatorname{Ker} \phi$?
97139
Let $R \simeq S$ be isomorphic commutative rings with unity. Prove the following:  a). If $R$ is an integral domain then $S$ is an integral domain.  For this, I said that If I let $f: R \to S$ be an isomorphism and $(R, + , \circ), (S, \oplus, \ast)$, then let $x, y\in R$,  such that $x \circ y = 0_R$, because $f$ is an isomorphism, $f(x \circ y) = f(x) \ast f(y) = f(0_R)$. If I let $R$ be an integral domain then, as per the definition $R$ is commutative, has unity, and no zero divisors.   So $f(x \circ y) = f(x) \ast f(y) = f(0_R)$ where we have either $f(x \circ y) = f(0_R) \ast f(y) = f(0_R)$ or $f(x \circ y) = f(x) \ast f(0_R) = f(0_R)$.   Since $f$ is an isomomorphism, let $s = f(x), t = f(y)| s,t \in S$. Then $s \ast t = f(0_R)$. By definition, $S$ also has unity so $0_S$ exists. If we map $0_R \to 0_S$. Then either $s = 0_S$ or $t = 0_S$.   So then, since $S$ has unity, is commutative, and has a no zero divisors as we showed above, it is also an Integral domain.  I'm not sure I have this correctly.  b). If $R$ is a field then $S$ is a field. I don't really understand how to do this one. Since $S$ is an integral domain, it has a $0_S$. So I need to show that $S$ distributes $\ast$ over $\oplus$.  Any help with these is appreciated
67393
Is $\mathbb{Z}[x]/(x^2+1)$  isomorphic to  $\mathbb{Z}[i]$?   My attempt is that try to define a mapping $g$ from $\mathbb{Z}[x]$ to $\mathbb{Z}[i]$ by $g(f(x))= f(i)$, for $f(x)\in\mathbb{Z}[x]$. If it is possible then $\ker g$ is $(x^2+1)$? Am I on the right track? Please Help.
28582
Hi I don't know how to show that $\langle 2,x \rangle$ is not principal and the definition of a principal ideal is unclear to me. I need help on this, please.   The ring that I am talking about is $\mathbb{Z}[x]$ so $\langle 2,x \rangle$ refers to $2g(x) + xf(x)$ where $g(x)$, $f(x)$ belongs to $\mathbb{Z}[x]$.
25570
I want to prove that the kernel of the evaluation map $s_a : \mathbb{C}[x_1,\dots,x_n] \rightarrow \mathbb{C}, x_i \mapsto a_i$ where $a = (a_1,\dots, a_n) \in \mathbb{C}^n$ is the ideal generated by $\{x_1 - a_1, \dots, x_n - a_n\}$.    The proof in the book first shows it easily for the case $a = 0$, and says that by substitutions of variables $x_i' = x_i - a_i$, you can show it for the rest of the cases.  I know that $f \in \ker s_a$ iff $f(a) = 0$.  And $\phi : \mathbb{C}[x_1,\dots, x_n] \rightarrow \mathbb{C}[x_1 - a_1, \dots, x_n-a_n], x_i \mapsto x_i - a_i$ is a surjective ring homomorphism.  Please give a hint and not the full answer.  Thanks.
64265
Are $\mathbb{C} \otimes _\mathbb{R} \mathbb{C}$ and $\mathbb{C} \otimes _\mathbb{C} \mathbb{C}$ isomorphic as $\mathbb{R}$-vector spaces?   I am having a very hard time at digesting tensor products and I do not know how to "compare" tensor products over different rings. My hunch is that they are not isomorphic. It is easy to see that $\mathbb{C} \otimes _\mathbb{R} \mathbb{C}$ is an $\mathbb{R}$-vector space of dimension $4$. I suspect that $\mathbb{C} \otimes _\mathbb{C} \mathbb{C}$ is also an $\mathbb{R}$-vector space but of lower dimension, but I have no idea how to show this or if indeed this intuition is correct.  Thank you.
81342
I came over a question in ring theory which I am not being able to proceed upon:     When is the ring $m\mathbb{Z}$ isomorphic to the ring $n\mathbb{Z}$, where $m, n \in \mathbb{N}$?   I know that to show isomorphism, I need to show that it is onto and the kernel consists of only $\{0\}_{m\mathbb{Z}}$ but I am not being able to write things down properly. Will someone help? What is the relation between $m$ and $n$?
112879
If $R$ is a ring and $I$ an ideal of this ring, is that true that $(R/I)[x]=R[x]/I$?   Note that can $I$ be identified as a subset of $R[x]$ and we need that in order to $R[x]/I$ makes sense.  I think it's true because every polynomial $(a_0+I)+(a_1+I)x,\ldots,(a_n+I)x^n$ of $(R/I)[x]$ can be written as a polynomial of $R[x]/I$ and vice versa since we have this equality: $$(a_0+I)+(a_1+I)x,\ldots,(a_n+I)x^n=(a_0+a_1x+\ldots a_nx^n) +I$$  Is my reasoning right?  Thanks
19185
Let $R$ be a commutative ring (with identity).   Let $f,g\in R\left[x\right]$ both be monic polynomials of degree $d$.   Then the underlying abelian groups of the rings $R\left[x\right]/\left(f\left(x\right)\right)$ and $R\left(x\right)/\left(g\left(x\right)\right)$ are isomorphic. Actually both are isomorphic to the direct sum $R^{\oplus d}$.  Questions:     What sort of relation between $f$ and $g$ will ensure that $R\left[x\right]/\left(f\left(x\right)\right)$   and $R\left(x\right)/\left(g\left(x\right)\right)$ are also isomorphic   as rings (with identities)?      What sort of relation between $f$ and $g$ will ensure that $R\left[x\right]/\left(f\left(x\right)\right)$   and $R\left(x\right)/\left(g\left(x\right)\right)$ are also isomorphic as   $R$-algebras?   Maybe my questions are too broad. Partial answers (e.g. with extra conditions on $R$) or references are also very welcome.
7149
This is a very simple question but I believe it's nontrivial.  I would like to know if the following is true:      If $R$ and $S$ are rings and $R[x]$ and $S[x]$ are isomorphic as rings, then $R$ and $S$ are isomorphic.    Thanks!  If there isn't a proof (or disproof) of the general result, I would be interested to know if there are particular cases when this claim is true.
81674
How do I prove that a finite ring $R$ of order 10 is isomorphic to the ring $\mathbb{Z}/10 \mathbb{Z}$?    I know that as a group under addition, $(R,+)$ is isomorphic to the group $(\mathbb{Z}/10 \mathbb{Z}, + )$, but the multiplication is rather mysterious to me.
89200
Let $R$ be a commutative ring and $A=\{1,a,a^2,\dots\}$ for some $a\in R$. Prove that $A^{-1}R$ is isomorphic to $R[T]/(aT-1)$.    I guess I'm meant to find a surjective homomorphism between $A^{-1}R$ and $R[T]$ and then use first isomorphism theorem. What homomorphism should I use?
73025
So I want to find the isomorphism $\phi$ that takes $F = \mathbb{Z}_3/\langle x^3 - x - 1\rangle$ to $E = \mathbb{Z}_3/\langle x^3 - x + 1\rangle$.  I understand that these are both finite fields of size 3^3 since $x^3 - x - 1$ and $x^3 - x + 1$ are both irreducible polynomials of degree 3.  I feel like a good start would be to take the subfield $\mathbb{Z_3}$ in $F$ to the subfield $\mathbb{Z_3}$ in $E$.  Or maybe find the right generator of each multiplicative group of size $3^3-1$ of each field and send those to each other.
109078
Im trying to show that $\mathrm{End}(\langle \mathbb{Z},+\rangle)$ is naturally isomorphic to $\langle \mathbb{Z},+,\cdot\rangle$, but I'm not quite sure which ring homomorphism to use.  Thank you
12199
I've a basic question.      $R$ is a simple left artinian ring. I want to show that all simple left $R$-modules are isomorphic.    A simple $R$-module $M$ is isomorphic to $R/J$ where $J$ is a maximal left ideal of $R$. Does this imply that $M\cong Ra$ for some $a\in R$, $a\neq 0$?   Many thanks.
47964
Let $R$ be the ring of functions that are polynomials in $\cos t$ and $\sin t$ with real coefficients.    Prove that $R$ is isomorphic to $\mathbb R[x,y]/(x^2+y^2-1)$.  Prove that $R$ is not a unique factorization domain. Prove that $S=\mathbb C[x,y]/(x^2+y^2-1)$ is a principal ideal domain and hence a unique factorization domain.  Determine the units of the rings $S$ and $R$. (Hint: Show that $S$ is isomorphic to the Laurent polynomial ring $\mathbb C[u,u^{-1}]$.)
32824
I'm having troubles with one of the problems in the book Introduction to Commutative Algebra by Atiyah and MacDonald. It's on page 11, and is the last part of the second question.  Given $R$ a commutative ring with unit. Let $R[x]$ be the ring of polynomials in an indeterminate $x$ with coefficients in $R$. We say that a polynomial $f = \sum\limits_{i=0}^n r_ix^i \in R[x]$ (with coefficients $r_0, r_1, \ldots, r_n$) is primitive if $\langle r_0,r_1,...,r_n\rangle = R$, i.e., the ideal generated by the coefficients of $f$ is $R$. Let $f$ and $g$ be two polynomials in $R[x]$. Prove that $fg$ is primitive iff $f$ and $g$ are both primitive.  The $\Rightarrow$ part is easy. Say, $f = \sum\limits_{i=0}^n r_ix^i$, $g = \sum\limits_{i=0}^m s_ix^i$; then $fg = \sum\limits_{i=0}^{m+n} c_ix^i$, where $c_k = \sum\limits_{i + j = k}r_is_j$. Since $fg$ is primitive, there exists a set of $\{\alpha_i\} \subset R$, such that $\sum\limits_{i=0}^{m+n} \alpha_ic_i = 1$, to prove $f$ is primitive, I just need to write all $c_i$'s in terms of $r_i$'s, and $s_j$'s, then group all $r_i$ accordingly, rearranging  it a little bit, and everything is perfectly done. And the proof of the primitivity of $g$ is basically the same. The $\Leftarrow$ part is just so difficult. Say $f = \sum\limits_{i=0}^n r_ix^i$, $g = \sum\limits_{i=0}^m s_ix^i$ are both primitive, then there exists $\{\alpha_i\}; \{\beta_i\} \subset R$, such that $\sum\limits_{i=0}^{n} \alpha_ir_i = 1$, and $\sum\limits_{i=0}^{m} \beta_is_i = 1$. At first, I thought of multiplying the two together, but it just didn't work. So, I'm stuck since I cannot see any way other than multiplying the two sums together. I hope you guys can give me a small push on this. Thanks very much in advance, And have a good day.