This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

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What are some good books to study Non -Commutative Rings?

What are some good books to study Non -Commutative Rings? I want to study structure of semisimple rings and Wedderburn -Artin Theorem in particular . The book should provide motivations and have ...
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0answers
30 views

Lasker-Noether Theorem and Kummer-Dedekind

I would like to know about the relations between Ernst Kummer's invention of complex ideal numbers (and Dedekind's development of them into what is now called ideals) regarding the unique ...
3
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2answers
111 views

Show that $\mathbb{C}[x,y]/(x^2+y^2-1)$ is a UFD. [duplicate]

I am trying to prove that the ring $\mathbb{C}[x,y]/(x^2+y^2-1)$ is a UFD. I have an hint, that suggests to find an isomorphism between $\mathbb{C}[x,y]/(x^2+y^2-1)$ and $\mathbb{C}[e^{it},e^{-it}]$, ...
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2answers
83 views

Show that, if $ a + bi$ is prime in $\mathbb{Z} [i]$, then $a - bi$ is prime in $\mathbb{Z}[i]$

Show that, if $ a + bi$ is prime in $\mathbb{Z} [i]$, then $a - bi $ is prime in $\mathbb{Z}[i]$ Since $\mathbb{Z}[i]$ is $ED$, then if $a+bi$ is irreducible then $a+bi$ is prime. But now how I ...
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1answer
22 views

Ring Homomorphisms from $\mathbb{Z} \to \mathbb{Z}/30\mathbb{Z}$

I am confusing myself here. Also, is it always understood that we are considering unital ring homomorphisms? $\phi(r)=\phi(\sum_1^r 1)= \sum_{i=1}^r \phi(1) = r \phi(1) = r(0+30 \mathbb{Z}) = r ...
3
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3answers
44 views

If $I$ and $J$ are ideals in $R$, and $I$ is a subset of $J$, is $I$ also an ideal in $J$?

Pretty much what the title suggests. If $I \subset J$ are both ideals in a commutative ring $R$, is it true that $I$ is an ideal in $J$? My reasoning for this is that clearly for all $a,b\in I$, $a ...
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1answer
25 views

Tensor product of division rings is Noetherian

Let $k$ be a field and $D_{1},D_{2}$ division rings which are finite dimensional over $k$. Is it true that $D_{1} \otimes_{k} D_{2}$ is Noetherian? Can we say that yes since the tensor product is ...
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1answer
24 views

If I is an ideal of $R$ ($R$ a ring) prove that $M_{n}(R/I)$ is isomorphic to $M_{n}(R)/M_{n}(I)$

If $I$ is an ideal of $R$ ($R$ a ring) prove that $M_{n}(R/I)$ is isomorphic to $M_{n}(R)/M_{n}(I)$ I proved that $M_{n}(I)$ is an ideal of $M_{n}(R)$ but I don't know how to prove this. Thanks for ...
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2answers
23 views

Let $A,B\in \Bbb{K}^{n^2}, B\neq0$ prove that if $AB=0$ then $A$ is not invertible.

Let $A,B\in \Bbb{K}^{n^2}, B\neq0$ prove that if $AB=0$ then $A$ is not invertible. I'm having trouble proving this, I tried saying that $|AB|=|A||B|=0 \implies |A|=0 \text{ or } |B|=0$ but that got ...
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1answer
53 views

Every subring of a field is a domain. Is this reciprocal?

I'm reading my notes on ring theory, and we proved on class that every subring of a field is a domain. Proof: Let $S \subseteq K$ be a subring of $K$, with $K$ a field. Let $x,y \in S$. If $xy=0$, ...
3
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1answer
44 views

Commutative Hereditary Rings

Is it true that the ring $\mathbb Z/n\mathbb Z$ ($n≠0$) is hereditary if and only if $n$ is square-free? The "if" part is OK to me because any field $\mathbb Z/p \mathbb Z$ ($p$ prime) is a PID, ...
4
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0answers
40 views

Definitions of the ring $R/(x_0^\infty,\ldots,x_{n-1}^\infty)$

Let $R$ be a ring, and let $I = (x_0,\ldots,x_{n-1})$ be a finitely-generated ideal inside of $R$, generated by a regular sequence. In algebraic topology one often encounters a ring, usually denoted ...
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48 views

Is an irreducible ideal in $R$ irreducible in $R[x]$?

Let $R$ be a commutative Noetherian ring and $I\subset R$ an ideal that is irreducible in the sense that if $I = J_1 \cap J_2$, then $I=J_1$ or $I=J_2$. Is (the ideal generated by) $I$ irreducible in ...
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0answers
36 views

Determine all ring homomorphisms from $ \Bbb Z$ $\oplus$ $\Bbb Z $ to $\Bbb Z$. [duplicate]

I got $(a,b) \to a$, $(a,b) \to b$ and $(a,b) \to 0$ these mappings to be homomorphisms just by hit and trial. So when I looked for it's solution, these were the ONLY homomorphisms from $ \Bbb Z$ ...
2
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1answer
28 views

$A\subseteq B$, $B$ integral over $A$, $\mathfrak{q}_{1}\subsetneq\mathfrak{q}_{2}$, then $A\cap\mathfrak{q}_{1}\subsetneq A\cap\mathfrak{q}_{2}$.

Let $A, B$ be commutative rings such that $A\subseteq B$ and $B$ is integral over $A$. I want to prove that if $\mathfrak{q}_{1},\mathfrak{q}_{2}$ are prime ideals of $B$ such that ...
2
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1answer
34 views

Show that R/P has only two elements .

Let $R$ be a Boolean ring and $ P $ be a prime ideal of $R. $Show that $R/P $ has only two elements . Then conclude that $ P $ is a maximal ideal. I start with $x^2 =x $ for all $x$ belong to $R$ . ...
2
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1answer
31 views

If $\overline{k}$ is an algebraic closure of a field $k$, then $\overline{k}[x_{1}, \dots, x_{n}]$ is integral over $k[x_{1}, \dots, x_{n}]$.

I want to prove that if $\overline{k}$ is an algebraic closure of a field $k$, then $\overline{k}[x_{1}, \dots, x_{n}]$ is integral over $k[x_{1}, \dots, x_{n}]$. It is used in exercise 11.3 of the ...
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1answer
18 views

Using Exchange Lemma in a decomposition

The "Exchange Lemma" in the decomposition of modules states that any decomposition of right $R$-modules $M_1⊕...⊕M_n=A⊕B$ with the endomorphism ring $End (A_R)$ local yields $M_i≈A⊕X$ for some $i$, ...
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3answers
57 views

Let $ I $ be an ideal in $\mathbb Z [i]$. Show that $\mathbb Z[i] /I $ is finite.

Let $I$ be an ideal in $\mathbb Z[i]$. I want to show that $\mathbb Z[i]/I$ is finite. I start with $Z[i]/I$ is isomorphic to $Z$. $Z$ is ID then $I$ is prime .Here i get stuck. Thanks for ...
3
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2answers
113 views

Does $IJ=IK\implies J=K$ always hold for integral domain and finitely generated nonzero ideal $I$?

Let $R$ be a commutative integral domain, $I,J,K$ three ideals of $R$ with $I\neq (0)$ being finitely generated. Then does $IJ=IK$ imply $J=K$? With Nakayama lemma, I can prove it if one of $J$ and ...
2
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1answer
32 views

Unique homomorphism between quotients

I am working on an exercise I found rather entertaining, albeit I found myself struggling at how to attack this problem as I don't know from which angle to approch it and tips or tricks would be ...
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0answers
66 views

Local ring at generic point

Let $X$ be a smooth projective variety, and $Y$ a subvariety of codimension one (both are irreducible). I want to show that the local ring $\mathcal{O}_{Y,X}$ at the subvariety $Y$ (which is nothing ...
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0answers
26 views

Proving that for an integral domain $R$, $y\in (x)\iff (y)\subseteq (x)$.

I am trying to prove the following statement. Let $R$ be a integral domain. Then for all $x,y\in R$ we have $$x\mid y\iff y\in(x)\iff (y)\subseteq (x).$$ Note that $(x)$ denotes the principal ...
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0answers
37 views

The opposite of the right ideal in the ring of 2x2 matrices?

Since every ring has an opposite, I would like to know: Which is the opposite of the rings of $n \times n$ matrices? More specifically, of the $2 \times 2$ matrices. Is there an opposite for the ...
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1answer
17 views

Ideals of $M_2(Q)$ [duplicate]

Could any one tell me how to show that only ideals of above ring are $(0)$ and the whole ring? My thought was to show that by taking any proper ideal if we can prove identity element belongs to it ...
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0answers
45 views

Ring epimorphism

If we have a ring homomorphism $f:X\rightarrow\mathbb Z$, where $X$ is the set of $2\times 2$ matrices of the form $$A=\left(\begin{array}{cc}x & y\\ y & x\end{array}\right),$$ defined by ...
0
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1answer
56 views

Union of specific prime ideals is not an ideal

Let $R$ be a commutative ring with $1$ with three prime ideals $P_1,P_2,P_3$ such that $P_i\subseteq P_j$ if and only if $i=j$. I want to show that the union of these prime ideals, which I denote ...
3
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2answers
88 views

For what kind of $R$-modules $M$ can we find an element $m\in M$ satisfing that $i:M\to M\otimes_R M, x\mapsto x\otimes m$ is an epimorphism?

Let $R$ be a commutative ring with identity and $M$ a $R$-module. I'm interested in under what condition we can find an element $m\in M$ satisfing that $i:M\to M\otimes_R M, x\mapsto x\otimes m$ is an ...
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1answer
40 views

Show that $R$ is an integral domain iff for all $ x, y, z\in R, xy = xz $ implies $y = z$.

Suppose $R$ is a commutative ring. Show that $R$ is an integral domain iff for all $ x, y, z\in R, xy = xz $ implies $y = z$. Proof: $\Rightarrow $Let $x,y,z\in R$ such that $x(y-z)=0$ ...
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1answer
48 views

Finding a primitive fifth root of unity modulo $81$ using a specific method.

I want to find a fifth root of unity modulo $81$ using a suggested method from the book (I can't come up with any other good method anyway). It is given that $x^4+x+2 \in \mathbb{F}_3[x]$ is ...
2
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2answers
55 views

Does $\text{End}_R(I)=R$ always hold when $R$ is an integrally closed domain?

Let $R$ be a commutative ring with identity, and $I\neq 0$ an ideal of $R$, I'm thinking how to calculate $\text{End}_R(I)$. I have proved that when $R$ is a integral domain, ...
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0answers
88 views

Study of rings of the form $R+I$

In my life I saw lots of ways of constructing rings: polynomial rings, quotient rings, localizations, endomorphism rings, rings of fractions, integral closure of a ring, center of a ring, etc... These ...
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0answers
34 views

Considering the prime ideals of $\Bbb Z$ to find the prime ideals of $\Bbb Z[x]$

Why can we consider the prime ideals of $\Bbb Z$ to determine the prime ideals of $\Bbb Z[x]$? There really isn't any work I can show here. Motivation is that it seems various papers consider a prime ...
2
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1answer
41 views

$\Bbb Z[x]/(p)\cong \Bbb F_p[x]$

How do I prove this? $$\Bbb Z[x]/(p)\cong \Bbb F_p[x]$$ This implies that $(p)$ must be equal to $p\Bbb Z$ where $p$ is prime. Why is the only prime element that makes this true $p$ where $p$ is ...
7
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1answer
75 views

Does $M_n(R_1)\cong M_n(R_2)$ imply $R_1\cong R_2$?

Let $R_1,R_2$ be two rings with identity. If for some $n\in\mathbb N$, $M_n(R_1)$ and $M_n(R_2)$ are isomorphic as rings, can we deduce that $R_1\cong R_2$? I can prove it when both $R_1,R_2$ are ...
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1answer
29 views

Finding explicit discrete valuation of ring of germs of analytic functions on $\mathbb{C}$

I found interesting problem set http://www.math.lsa.umich.edu/~kesmith/593hmwk2-2014.pdf and I noted Problem 3-3. And I found another version: Let $\mathcal{U}$ be the subset of all open sets of ...
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0answers
37 views

Understanding a quotient ring of continuous functions

In trying to understand another questions answer(to a question I asked), I realized that my fundamental lack of knowledge was in regards to the following question: In terms of functions, what does ...
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2answers
55 views

Maximal ideals of the ring of all continuous functions [duplicate]

Let $R$ be the ring of all continuous functions from the interval $[0,1]$ to $\Bbb R$. For each $a\in[0,1]$ let $$A_a=\{f\in R \mid f(a)=0\}$$ Now firstly, this is part of an assignment problem, ...
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1answer
45 views

What does it mean for a prime ideal to split completely?

See here. What does it mean for a prime ideal to split completely?
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1answer
67 views

Extension $\mathbb{Q}(\sqrt{-5}, i)/\mathbb{Q}(\sqrt{-5})$, splitting.

In the extension $\mathbb{Q}(\sqrt{-5}, i)/\mathbb{Q}(\sqrt{-5})$, must principal prime ideals of $\mathbb{Z}[\sqrt{-5}]$ necessarily split into 2? Must nonprincipal prime ideals not split? ...
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2answers
29 views

irreducibility of polynomials made by perturbation from a polynomial

Suppose $f(x)\in\mathbb{Z}[x]$ with $\text{deg}f=2n,n\in\mathbb{Z_+}$ and $f_m(x):=f(x)+ mx^n $ for each integer $m\in\mathbb{Z}$. Let us define a number $P_f$: ...
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1answer
72 views

How to prove that the ring of algebraic integers is a Bézout domain?

I was told that the ring of all algebraic integers (that is, the complex numbers which are roots of a monic polynomials with integral coefficients) is a Bézout domain. But I have no idea how to prove ...
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3answers
53 views

Show that $f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n\in R[x]$ is nilpotent iff $a_0,a_1,a_2,\ldots,a_n$ are nilpotent

Let $R$ be a commutative ring. Show that $f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n\in R[x]$ is nilpotent if and only if $a_0,a_1,a_2,\ldots,a_n$ are nilpotent. Now since $f(x)$ is nilpotent then ...
1
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1answer
58 views

Looking for a direct proof that all maximal ideals of $\mathbb C[x_1,x_2,…,x_n]$ are generated by $n$ linear polynomials

Without using Hilbert's Nullstelensatz , can we directly prove that all maximal ideals of $\mathbb C[x_1,x_2,...,x_n]$ is of the form $\langle x-a_1,x-a_2,...,x-a_n \rangle$ ? It is easy to prove it ...
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2answers
28 views

Show that if $\phi(F) \neq \{0\}$ then $F \cong R$.

Let $F$ be a field. Let $R$ be a ring and suppose $\phi : F \rightarrow R$ is an onto ring homomorphism. Show that if $\phi(F) \neq \{0\}$ then $F \cong R$. (Prove $F$ isomorphic to $F/\{0\}$ first) ...
2
votes
1answer
76 views

Number of ideals in a minimal irreducible decomposition

Assume $R$ is a local ring, $M\subseteq R$ is the maximal ideal, $I\subseteq R$ is an $M$-primary ideal and $I=\bigcap_{i=1}^n Q_i$ is a minimal irreducible decomposition of $I$ (i.e. $Q_i\subseteq R$ ...
2
votes
2answers
75 views

How to define a specific ring using a homomorphism

If we have a ring $R$ then I can form a ring of matrices isomorphic to $R$ by setting $r \overset{\phi}{\mapsto} \left( \begin{array}{ccc} r & 0 \\ 0 & 0 \end{array} \right) $ and defining ...
12
votes
2answers
77 views

What is $\mathbb{Z}[i] \otimes_{\mathbb{Z}[2i]} \mathbb{Z}[i]$?

What is $\mathbb{Z}[i] \otimes_{\mathbb{Z}[2i]} \mathbb{Z}[i]$? Also, since $\mathbb{Z}[i]$ is a PID, we should be able to write this $\mathbb{Z}[i]$-module as a direct sum of cyclic ...
7
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2answers
132 views

For any rng $R$, can we attach a unity?

Let $R$ be an rng. (There may be no unity) Then, does there always exist a ring(with unity) $A$ such that $R$ is a subrng of $A$?
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0answers
38 views

Is there a finite ring whose rank is smaller than the rank of its group and its monoid?

Consider a finite ring $(R, +, \times)$ comprising a finite additive abelian group $(R, +)$, a finite multiplicative monoid $(R, \times)$, and a distributivity rule relating the two. Let the rank of ...