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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0
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1answer
29 views

$K\leq_s M\leq N$. Is it true that $K\leq_s N$?

I've got another question. I'm currently studying the section concerning small (inessential or superflous, as you wish) submodules: namely we define $$N\leq_s M$$ if $N\leq M$ and whenever $L\leq M$ ...
1
vote
1answer
59 views

prove that $R=e_1R\times e_2R\times\cdots\times e_nR$

I have troubles in showing this fact. Let $R$ be a ring, and assume that $e_1,\dots,e_n$ are central idempotents elements of $R$, i.e. idempotents contained in the center of $R$. Assume moreover that ...
3
votes
3answers
119 views

Factorizing a polynomial $f$ in $A[x]$ (with $A$ commutative), where $f$ has a zero in its field of fractions

Let $A$ be a commutative ring and $S$ a multiplicative subset of $A$ generated by $s\in A$ which is not a zero-divisor. Consider the polynomial ring $A[x]$. Given a polynomial $f\in A[x]$, ...
3
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2answers
192 views

What is really the meaning of “$x$” in a ring of polynomial?

This question seems to be so easy, but it will be helpful to be clear especially for students. Some define $x$ as asequence like $(0,1_{R},0...)$ which $R$ is a ring and other regard $x$ as a variable ...
4
votes
1answer
124 views

Two isomorphisms of (algebraic) inverse limits

I am having trouble seeing why the following two isomorphisms should hold for a Noetherian ring $A$ and ideals $I$,$J$ of $A$: $$\varprojlim A/(I+J)^n \cong \varprojlim A/I^n+J^n$$ $$\varprojlim_m ...
4
votes
2answers
446 views

Left and right ideals of $R=\left\{\bigl(\begin{smallmatrix}a&b\\0&c \end{smallmatrix}\bigr) : a\in\mathbb Z, \ b,c\in\mathbb Q\right\}$

If $$R=\left\{ \begin{pmatrix} a &b\\ 0 & c \end{pmatrix} \ : \ a \in \mathbb{Z}, \ b,c \in \mathbb{Q}\right\} $$ under usual addition and multiplication, then what are the left and right ...
1
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2answers
169 views

If a sub-C*-algebra does not contain the unit, is it contained in a proper ideal?

Suppose that $R$ and $S$ are unital rings and that $S$ is a subring of $R$ in the weak sense where the multiplicative identities $1_R$ and $1_S$ are not assumed to be the same. In fact, assume $1_R ...
1
vote
4answers
410 views

Proper subring of $\mathbb{Z}_{12}$?

Can someone explain why there are no proper subrings of $\mathbb{Z}_{12}$? My explanation is that any proper subset of $\mathbb{Z}_{12}$ would have a different 0 element. Thus, it would not be a ...
2
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1answer
144 views

Hereditary Algebras

I recently began to study representation theory of algebras and I found this problem: Suppose $\Lambda$ is a finite dimensional algebra over an arbitrary field. If $\Lambda$ is hereditary, basic ...
3
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1answer
93 views

G additive group isomorphic to $\mathbf{Z}^{n}, \mathbf{Z}^{m}$

I have been stuck with this problem for quite a while now: G is an additive group with $n,m \in \mathbf{N}$ ; $\mathbf{Z}^{n}, \mathbf{Z}^{m}$ isomorphic. Then it holds that: $\#(G/2G) = 2^{n} = ...
1
vote
3answers
267 views

$I$ prime does not imply $I$ maximal

I am attempting to prove the converse to $I$ maximal implies $I$ prime is not true. $I$ prime $\iff A/I$ is an integral domain and $I$ maximal $\iff A/I$ is a field so I'm looking for a prime ideal ...
1
vote
2answers
109 views

Isomorphism on a collection of R, $\mathbf{Z}$-isomorphism between ideal and $\mathbf{Z}[5i]$

tough (*) question from a previous examination paper: 6.(*) a) Let $R=\mathbf{C}[q]/q^{2}\mathbf{C}[q]$ and let $m,n \in \mathbf{N}$ where $R^{m},R^{n}$ are R-isomorphic. Show that m is equal ...
2
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1answer
84 views

Determination of the number of $R$ submodules where $R= \mathbb{Z}/4\mathbb{Z}$

Suppose $R=\mathbb{Z}/4\mathbb{Z}$. i) How many R-submodules $M= Rx \subset R^{2} \ (x\in R^{2}) $ are there? ii) How many equivalence classes are isomorphic to M? i) definition of a submodule: ...
2
votes
2answers
107 views

factorial ring basis $\mathbb{Z}/2\mathbb{Z}$

i) $\mathbb{Z}/2\mathbb{Z}$ has no $\mathbb{Z}$ basis. ii) $\mathbb{Z}/2\mathbb{Z}$ has no $\mathbb{Z/2\mathbb{Z}}$ basis. iii) Suppose $R= \mathbb{Z}[t]$. Then $2R+tR$ has no R-basis. i) ...
2
votes
1answer
140 views

$\mathbb{C}[t]$ module , module isomorphisms

If $V$ is a $\mathbb{C}$ vector space and $a\in End \ V $. Then let $V=V_{a}$ be the $\mathbb{C}[t]$-module defined by:$$Pv=P(a)v, \ \ (P\in \mathbb{C}[t], \ v\in V)$$ i) Suppose $M$ is a ...
5
votes
1answer
219 views

Applications Wedderburn's Little Theorem

I've been asked to give a short presentation of Wedderburn's Theorem that every finite domain is a field. However, the proof itself is quite short so I thought to add some applications (since this ...
1
vote
1answer
102 views

Dimension of the vector space? $M_n(K)$?

How do you calculate the dimensions of a vector space more generally. For any field $K$ and $n \in \mathbb{N}$, $M_{n}(K)$ is an algebra over K. The notes says that the vector space dimension is ...
0
votes
1answer
35 views

Why is $ax=(a 1_{a})x$ for $x \in A$, $a \in K$ a K-algebra?

Definition. Let K be a field. An algebra A over K is a set on which three operations are defined, addition, multiplication and multiplication by a scalar such that i) A is a ring under addition and ...
2
votes
1answer
67 views

Operations on generators of an ideal

Suppose I have an ideal $I = (r, s, t)$. What operations can I apply to the $r$, $s$, and $t$, so that I get an ideal equal to $I$? For example, $(r, s, t) = (r - s, s, t)$. What are the other ones?
2
votes
1answer
290 views

Can zero divisor in a non-associative ring be a unit?

Statement Left or right zero divisors in ring can never be units. is proved in Wikipedia this way: If $a$ is invertible and $ab = 0$ $0 = a^{−1}0 = a^{−1}ab = b$ I'm confused by ...
1
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4answers
171 views

If $\pi:\mathbb{Z}\to \mathbb{F}$ so that $\pi(0)=0$ and $\pi(n+1)=\pi(n)+1$ why is the image a subfield?

Suppose $\mathbb{F}$ is a field. Define $\pi:\mathbb{Z}\to \mathbb{F}$ so that $\pi(0)=0$ and $\pi(n+1)=\pi(n)+1$ for all $n \geq 0$. If $n<0$ then $\pi(n)=-\pi(-n)$. So $\pi$ is a homomorphism. ...
4
votes
2answers
2k views

Proving $2,3,1+\sqrt{-5}$ and $1-\sqrt{-5}$ are irreducible in $\mathbb{Z}[\sqrt{-5}]$

Could anyone help me prove that $2,3,1+\sqrt{-5}$ and $1-\sqrt{-5}$ are irreducible in $\mathbb{Z}[\sqrt{-5}]$? As $6=2*3=(1+\sqrt{-5})(1-\sqrt{-5})$ so $\mathbb{Z}[\sqrt{-5}]$ is not a UFD. ...
2
votes
1answer
63 views

Show that $\text{Hom}(eR,-)$ and $-e$ are naturally isomorphic functors

Good Morning everyone. I'm currently having trouble with the following: Problem: Show that, if $e$ is an idempotent element of a ring $R$, then the two functors ...
2
votes
2answers
763 views

How to find the GCD of two polynomials

How do we find the GCD $G$ of two polynomials, $P_1$ and $P_2$ in a given ring (for example $\mathbf{F}_5[x]$)? Then how do we find polynomials $a,b\in \mathbf{F}_5[x]$ so that $P_1a+ P_2b=G$? An ...
2
votes
3answers
431 views

A ring with a subring that is a field

I'm looking for an example of a ring $R$ such that $R$ has no multiplicative identity, but R has a subring $A$ which is a field
4
votes
2answers
231 views

How do you show that $\mathbb{Z}[x]/(x^2-29) \cong \{a+b\sqrt{29}|a,b \in \mathbb{Z}\}$?

$\mathbb{Z}[x]/(x^2-29) \cong \{a+b\sqrt{29}|a,b \in \mathbb{Z}\}$. I understand what $\mathbb{Z}[x]/(x^2-29)$ means: that every polynomial that can be factored by $(x^2-29)$ is equal to zero. ...
0
votes
1answer
185 views

Jordan-Hölder factors of a finite length module

Suppose one has a local ring $(A,\mathfrak{m})$ and a finite length $A$-module $M$ with $\operatorname{supp}(M) = \{\mathfrak{m}\}$. Does $M$ have a composition series consisting only of ...
2
votes
2answers
112 views

What is $\operatorname{End}_{\mathbb{C}}(\mathbb{C}[x])?$

Can someone explain what $\operatorname{End}_{\mathbb{C}}(\mathbb{C}[x])$ is? I just want to know what its elements look like. In the definition, it says that for a field $K$ and $n \in \mathbb{N}$, ...
0
votes
2answers
148 views

How is $K[X]$ an algebra over K?

From notes an algebra A over K. Has three properties i) A is an ring under addition and multiplication. ii) A is a vector space over K under addition and scalar multiplication. iii) for all $\alpha ...
2
votes
1answer
103 views

Module homomorphisms $\prod \mathbb{Z} \rightarrow \mathbb{Z}$

Let $f\colon \prod \mathbb{Z} \rightarrow \mathbb{Z}$ be a $\mathbb{Z}$-module homomorphism. I want to show that $f(e_i) = 0$ for almost all $i$ where $e_i$ are the standard unit vectors. I assume ...
28
votes
4answers
1k views

Fermat's Last Theorem and Kummer's Objection

In 1847 Lamé had announced that he had proven Fermat's Last Theorem. This "proof" was based on the unique factorization in $\mathbb{Z}[e^{2\pi i/p}]$. However, Kummer, proved that when $p=23$ we do ...
4
votes
1answer
291 views

Proof for an integral domain involving subrings

Theorem: if $R,S$ are integral domains, $R\subset S$, and where $s_{1},...,s_{n}$ in $R_{S}$. Then there is a $m\in \mathbb{N}$ and $t_{1},..., t_{m}$ in $S$ (not all 0) so that $s_{i}N\subset N $ ...
4
votes
1answer
134 views

No linear polynomial combination in an ideal

tough (*) question from an old examination paper: $I=4R+2xR+x^{2}R$ is an ideal of $R= \mathbf{Z}[x]$. Show that there are no $U,V$ in $R$ , so that $I=UR+VR$ holds. Hardmath, Thank you! You ...
4
votes
2answers
634 views

Noetherian Ring under a Homomorphism / homomorphic function / map

Assume R is noetherian and $f:R \to S$ is a ring homomorphism. Is $f(R)$ noetherian? Reading Dylan Moreland commentary: "If $\mathfrak{b}$ is an ideal of $f(R)$, then $f^{-1}(\mathfrak{b})$ is ...
3
votes
1answer
118 views

Showing a ring is artinian?

We have a ring $R=\begin{pmatrix} \mathbb{Z} & 0 \\ \mathbb{Z} & \mathbb{Z} \end{pmatrix}$ Let $I=\begin{pmatrix} 12\mathbb{Z} & 0 \\ 3\mathbb{Z} & 3\mathbb{Z} \end{pmatrix}$ How do ...
4
votes
1answer
224 views

Ideal In Polynomial intersection of Ideal and integers

Let $I = AR + BR $ be an ideal of $R= \mathbb{Z}[x]$. i) Can you show that $I \cap \mathbb{Z} = t\mathbb{Z} ; t\in \ \mathbb{Z}$ ? ii) Can you show that if $t\ge 1$, then it holds that ...
2
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1answer
331 views

about the fractional ideal of a field of fractions

In the wikipedia article http://en.wikipedia.org/wiki/Fractional_ideal we read Let $R $ be an integral domain, and let $K$ be its field of fractions. A fractional ideal of $R$ is an $R$-submodule ...
3
votes
3answers
547 views

GCD in polynomial rings with coefficients in a field extension

Let $E|F $ be a field extension and $f,g$ $\in$ $F[x]$ (the polynomial ring with coefficients in $F$ ). Let's denote with $(f,g)_F$ the greatest common divisor of $f$ and $g$ in $F[x]$. Is it true ...
5
votes
2answers
372 views

Ring homomorphism between $R[x]$ and $R$

I am trying to show that, if $\phi : R[x] \rightarrow R$ is a ring homomorphism, such that $\phi$ restricted to $R$ is identity map, then $\phi=\phi_a$ for some $a$, where $\phi_a$ is the substitution ...
6
votes
1answer
502 views

Prime ideals of the ring of rational functions

Let $A$ be a commutative ring with identity. If $f = a_0 + a_1 x + \cdots + a_n x^n \in A[x]$ is a polynomial, define $c(f) = A a_0 + A a_1 + \cdots + A a_n$ the ideal of $A$ generated by the ...
3
votes
1answer
407 views

Proving a basic property of polynomial rings

I am learning ring theory in the Dummit & Foote's Abstract Algebra, and I am doing all the exercises to get as much experience as possible... but some of them just get me stuck for hours! Like ...
4
votes
3answers
2k views

A proof that this set is an ideal of a commutative ring

This is a homework problem which I have worked hard on, but got stuck at the last step. Any assistance would be much appreciated. The problem is from Herstein's Abstract Algebra, 3rd ed., section 4.3, ...
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2answers
500 views

Question about CRT

The question rephrased and compressed: Let $F=F_2[a]$ be a finite extension field of the field of two elements $F_2$. We are given a polynomial $R(X)\in F[X]$, and pairwise coprime irreducible ...
4
votes
2answers
972 views

Is $\mathbb Z[ \sqrt{-3}]$ UFD?

Is $\mathbb Z[ \sqrt{-3}]$ UFD ? If so , why $$4=2 \times 2 = (1+\sqrt{-3}) \times (1-\sqrt{-3} )$$ and every terms are irreducible ?
2
votes
1answer
245 views

Prove that a ring is commutative if $(ab)^2=(ba)^2$

Let $R$ be a ring with unit element $1$ such that $(ab)^2=(ba)^2$ for all $a,b$ in $R$. If in $R$, $2x=0$ implies $x=0$, how do I show that R is commutative? Is there any general approach to attack ...
6
votes
3answers
2k views

Zero divisor in $R[x]$

Let $R$ be commutative ring with no (nonzero) nilpotent elements. If $f(x) = a_0+a_1x+\cdots+a_nx^n$ in $R[x]$ is a zero divisor, how do I show there's an element $b \ne 0$ in $R$ such that ...
1
vote
2answers
184 views

Application of the Schönemann-Eisenstein criterion

Let $P\in \mathbb{Z}[x]$ be a polynomial of degree 2011. Can you show that there is an infinite amount of such polynomials, such that $P$ and $P+2$ remain irreducible? $x^{2011}+15ax+3$ is a ...
6
votes
3answers
225 views

When is a local algebra reduced?

Let $k$ be a field and let $A$ be a local $k$-algebra which has finite dimension over $k$. Let $\mathfrak{m}$ be the maximal ideal of $A$ and let $k' = A / \mathfrak{m}$ be the residue field. For ...
7
votes
1answer
367 views

Preservation of direct sums and finite generation

This is a follow up on Evariste's question Hom and direct sums. Let $R$ be an associative ring with one. The word "module" shall mean left $R$-module. Say that a module $A$ preserves direct sums if ...
1
vote
1answer
87 views

If $R$ contains two ideals $B$ and $C$ with $B+C=R$ and $B \cap C=0$, then $B$ and $C$ are rings and $R\cong B\times C$

If $R$ contains two ideals $B$ and $C$ with $B+C=R$ and $B \cap C=0$, then $B$ and $C$ are rings and $R\cong B\times C$ I tried to prove it using definition of subrings, that is, B and C is ...