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

learn more… | top users | synonyms (1)

2
votes
1answer
33 views

Irreducibles and factorization in $\mathbb{Z}[\sqrt{5}i]$

Consider the ring $\mathbb{Z}[\sqrt{5}i]=\{m+n\sqrt{5}i:m,n\in\mathbb{Z}\}$. Show that $21$ has two distinct factorisations into irreducibles in $\mathbb{Z}[\sqrt{5}i]$, which is thus not a UFD. ...
1
vote
2answers
48 views

$\Bbb{Z}/10\Bbb{Z}$ isomorphic to $\Bbb{Z}[i]/\langle 1+3i\rangle$.

I need to prove that $\Bbb{Z}/10\Bbb{Z}$ is isomorphic to $\Bbb{Z}[i]/\langle 1+3i\rangle$. I know I can use the third isomorphism, but I would like to use the first one. I consider a homomorphism ...
3
votes
3answers
84 views

Prove $I$ is non-principal ideal of $\mathbb{Z}[x]$? [duplicate]

I'm new to algebra and got stuck with concept of ideals. The question is to prove that $$I = \left\{ {{a_0} + {a_1}x + \cdots + {a_n}{x^n} \mid {a_i} \in \mathbb{Z},{a_0} \in 2\mathbb{Z}} ...
2
votes
0answers
36 views

Inverse limit of the following system.

Let $\mathcal{O}_L = \mathbb{Z}[\sqrt{-5}]$ be the number ring of $L=\mathbb{Q}[\sqrt{-5}]$, and $\mathfrak{p} =(2, 1+\sqrt{-5})$ a prime (and maximal ideal) in $\mathcal{O}_L$. What is the inverse ...
2
votes
0answers
35 views

When is the set $A=\{a+s|a\in I , s\in S \}$ a prime ideal of R?

Let $R$ be commutative ring with identity, $I$ an ideal of $R$, and $S$ a subset of $R$. Under what conditions is the set $A=\{a+s\mid a\in I , s\in S \}$: 1- an ideal of $R$? 2- a prime ...
0
votes
1answer
15 views

Length of a ring? Lenth of a (right or left) ideal

I have seen the concept of length being applied to rings. What is exactly mean by that? What does length mean in a statement like "the composition length of RR is 2, but the composition length of RR ...
0
votes
0answers
53 views

In what conditions every ideal is an extension ideal?

Let $R$ and $S$ be commutative rings (with $1$) and $f : R\to S$ be a ring homomorphism. For an ideal $I$ of $R$, set $I^e:=\langle f(I)S\rangle$ (called the extension of $I$ to $S$). When $f$ is ...
1
vote
0answers
43 views

Extensions of Vector Spaces

I have just finished learning about field extensions and such. In particular I am interested in the minimum polynomial of $a$ in some field extension $K$ of $F$. Particularly I have learned about the ...
0
votes
1answer
23 views

Tensor product of commutative rings

I need help with this question: Suppose that A, B, C are commutative rings with unit. Is it true that $A\otimes_\mathbb{Z}(B\times C)$ is isomorphic as rings with $(A\otimes_\mathbb{Z} ...
5
votes
5answers
162 views

Minimal Polynomial of $\sqrt{2}+\sqrt{3}+\sqrt{5}$

To find the above minimal polynomial, let $$x=\sqrt{2}+\sqrt{3}+\sqrt{5}$$ $$x^2=10+2\sqrt{6}+2\sqrt{10}+2\sqrt{15}$$ Subtracting 10 and squaring gives ...
0
votes
1answer
28 views

Generating functions which are prime

Sorry for strangely worded title. The intended meaning is the generating functions which are not divisible by other generating functions, not functions for generating prime numbers. With this out of ...
0
votes
2answers
19 views

prime ideals, ring

If $ab \in 6\mathbb{Z}$, it does not follow that $a$ or $b$ is in $6\mathbb{Z}$. For example, $2 \cdot 3 = 6 \in 6\mathbb{Z}$, but $2$ nor $3$ is in $6\mathbb{Z}$. Can someone explain why? ...
1
vote
2answers
78 views

Show that $RG \cong R \otimes_{\mathbb{Z}}\mathbb{Z}G$ [closed]

Let $G$ be a group and $R$ a commutative ring. Show that $$RG \cong R \otimes_{\mathbb{Z}}\mathbb{Z}G,$$ where $RG=\{\sum_{g \in G} a_gg ; a_g \in R\}$ and only a finite number of $a_g$ are nonzero.
15
votes
0answers
156 views

Can any commutative ring of characteristic $p\in\mathbb P$ be written as the form $R/(p)$ with $R$ being a ring of characteristic $0$?

Let $S$ be a commutative ring with identity with $\operatorname{char}S=p$, where $p$ is a prime number. I wonder if we can always find a ring $R$ such that $\operatorname{char}R=0$ and $R/(p)\cong ...
1
vote
0answers
30 views

Tensor products of ideals

Consider the class of complex algebras where the tensor products are over complex numbers. Given a complex algebra $A$ and a left ideal $L$ of $A$ generated by $n$ elements. Is $L^{\otimes n}$ ...
1
vote
1answer
44 views

If $p(x)\in F[x]$ is irreducible and $F$ has characteristic zero, then $p(x)$ has no repeated roots.

I want to prove that if $p(x)\in F[x]$, where $F$ is a field, is irreducible and $F$ has characteristic zero, then $p(x)$ has no repeated roots. I found this argument in a book, but I don't ...
2
votes
2answers
35 views

$(f(x))^p\neq f(x^p)$ on infinite field of characteristic $p$

It is easy to prove that if $f(x)\in\mathbb{Z}_p[x]$ then $(f(x))^p=f(x^p)$. Now, I want to show that this may be false if $\mathbb{Z}_p$ is replaced by an infinite field of characteristic $p$. The ...
5
votes
1answer
59 views

Separability of $A \subseteq C$ implies separability of $B \subseteq C$, where $A \subseteq B \subseteq C$

For commutative rings $R \subseteq S$, recall that $S$ is separable over $R$, if $S$ is a projective $S \otimes_R S$-module. (via $f: S \otimes_R S \to S$ given by: $f(s_1 \otimes_R s_2)=s_1s_2$). My ...
2
votes
1answer
43 views

Ring theory associates

Can someone please give me an example of of this definition, as I am finding it hard to get my head around or even understand what an "associate" is. Let $R$ be a commutative ring with unity. ...
1
vote
1answer
23 views

If $I$ is a maximal ideal in $R$, $(I,x)$ is a maximal ideal in $R[x]$

Click Link to Original Text Let $R$ be a commutative ring with $1$, and $I$ is an ideal of $R$. Then, $(I) = I[x]$ is an ideal in $R[x]$. I was able to prove, via first isomorphism, that ...
0
votes
1answer
31 views

Trying to prove that for a ring $R$ with identity, $R^2 = R$

I'm trying to prove that if a ring $R$ has an identity, then $R^2 = R$. Honesty I don't even know where to start. Any help would be appreciated. Thank you very much, Eric
0
votes
0answers
29 views

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 ...
1
vote
0answers
28 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
votes
2answers
100 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 $\mathbb{C}[e^{it},e^{-it}]$, ...
1
vote
2answers
80 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 ...
1
vote
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
votes
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 ...
-1
votes
1answer
23 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 ...
0
votes
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 ...
0
votes
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 ...
8
votes
1answer
52 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
votes
1answer
41 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
votes
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 ...
0
votes
0answers
47 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 ...
1
vote
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
votes
1answer
27 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
votes
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
votes
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 ...
0
votes
1answer
14 views

Using Exchange Lemma in an 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 $i$, ...
-2
votes
0answers
43 views

The set of elements being integral over a ring [closed]

Let $R$ be a ring. Let $A$ be a subring of $R$. An element $x\in R$ is called an integral element over $A$ if there exists a monic polynomial in $A[x]$ such that $f(x)=0$. Prove that the set of ...
3
votes
3answers
55 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
votes
2answers
112 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
votes
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 ...
1
vote
0answers
52 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 ...
0
votes
0answers
25 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 ...
0
votes
0answers
33 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 ...
0
votes
1answer
14 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 ...
0
votes
0answers
42 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
votes
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
votes
2answers
86 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 ...