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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2answers
53 views

Every element in a ring different from 0 is invertible [closed]

True or false.All number r in a ring R, different from 0 is same. It seems like it is true but how to go about proof if we consider Matrix then it is invertible if determinant is not 0 .
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0answers
52 views

Maximality of the kernel of a quasifield

Setting A (left) quasifield is an algebraic structure $(Q,+,\cdot)$ such that $(Q,+)$ is an abelian group. (As usual, we denote its identity element by $0$.) Each equation $x\cdot a = b$ with ...
1
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1answer
27 views

Find the remainder of $(p-2)!$ module $p$, where $p$ is a prime $\geq 3$

My attempt: From Wilson's Theorem: For a prime $p$, $$(p-1)! \equiv (-1) \pmod p$$ Multiplying both sides by $(p-2)$, $$(p-2)! \equiv -(p-2) \pmod p$$ i.e. $$(p-2)! \equiv 2 \pmod p$$ So the ...
9
votes
2answers
268 views

The ring of idempotents

Let $R$ be a commutative ring. Then its ring of idempotents $I(R)$ consists of the idempotent elements of $R$, with the same multiplication as in $R$, but with the new addition $x \oplus y := ...
0
votes
2answers
80 views

Find the remainder of $49!$ modulo $53$

Since $53$ is prime, from Wilson's theorem, $52! \equiv -1\pmod{53}$, i.e. $52 \times 51 \times 50 \times 49! \equiv -1\pmod {53}$ I don't understand how to take it from here. The other form I ...
0
votes
1answer
30 views

Solve $22x \equiv 5(mod 15)$

I looked at an example of this type, and here's my attempt: $gcd(22,15)=1$ and $1$ is a divisor of $5$ so solutions exist. Now $22x \equiv 5(mod 15)$ is the same as solving $22x=5$ in $Z_{15}$ i.e. ...
0
votes
1answer
47 views

Compute the remainder of $2^{(2^{17})}+1$ when divided by $19$

Compute the remainder of $2^{2^{17}}+1$ when divided by $19$ Hint given in book: Computer the remainder of $2^{17}$ modulo $18$ My attempt: From Fermat's little theorem, $2^{18}=1(mod19)$ I have ...
0
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1answer
39 views

Problem related to modules and k-algebras

I am trying to do the following exercise: Prove that if $A$ is a $k-$algebra and $M$ is a module then the product $\lambda * m=\tau(\lambda)m$ ($\tau$ is the morphism from $K$ to $Z(A)$) defines on ...
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2answers
102 views

Finding all primitive polynomials of a certain degree in $\mathbb{F}_q$

I am writing an algorithm to find all primitive polynomials in $\mathbb{F}_2[X]$ and I found this theorem : If $P(X)$ is a primitive polynomial in $\mathbb{F}_p[X]$ of degree $n$ with root $a$, then ...
0
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3answers
68 views

Find the characteristic of the ring $\mathbb Z_6 \times \mathbb Z_{15}$

My attempt: Let the characteristic be $n$. Then, $n \cdot (1_6, 1_{15}) = (0_6, 0_{15})$, i.e. $n \cdot 1_6=0_6$ and $n \cdot 1_{15}=0_{15}$ The least $n$ for which both are true is $30$, so $30$ ...
0
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1answer
56 views

Solve the equation $3x=2$ in the fields $\mathbb Z_7$ and $\mathbb Z_{23}$

This is a sum from Abstract Algebra by Fraleigh. Myy attempt: $$3x=2$$ $$\Rightarrow 3x-2=0$$ Now, the elements of $\mathbb Z_7$ are {$0,1,2,3,4,5,6$} Substituting these values in the left side ...
0
votes
1answer
42 views

A simple module

Suppose $R$ is a ring and $M$ is an $R$-module. Prove: $M$ is simple if and only if there is a left maximal ideal $m$ such that $M\cong R/m$.
1
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1answer
117 views

Nonzero Prime Ideals are Maximal in Euclidean Domains

Prove that every nonzero prime ideal in a Euclidean domain is maximal. This is what I have so far: Let $R$ be a euclidean domain and let $P$ be a nonzero prime ideal in $R$ generated by $a$. ...
2
votes
2answers
45 views

$|R|=30$ and $|I|=10$ then $I$ is a maximal ideal

How shall I check this: Suppose that $R$ is a commutative ring and $|R|=30.$ If $I$ is an ideal of $R$ and $|I|=10$ show that $I$ is a maximal ideal. Please give some hint how to go with solution...
1
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1answer
85 views

If $R$ is a ring and $A$ is a maximal ideal of $R$ then $R/A$ is a field

I've to prove one way : If $R$ is a ring and $A$ is a maximal ideal of $R$ then $R/A$ is a field. Now suppose that $A$ is maximal and let $b \in R$ but $b \notin A$. It suffices to show that $b+A$ ...
0
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1answer
18 views

A question about co-prime polynomials in $\Bbb{C}[x,y]$

Say $f$ and $g$ are two co-prime polynomials in $\Bbb{C}[x,y]$. Can the following expression always be written $$af+bg=1$$ where $a,b,f,g\in\Bbb{C}[x,y]$? I realise that the Euclidean algorithm is not ...
0
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1answer
47 views

writing elements of $\mathbb Z_5[x]/I$

If I have ring $\mathbb Z_5[x]$ ,and let $I=\langle x^2+x+2 \rangle$,then if I have to represent the elements of $\mathbb Z_5[x]/I$ they'll be of form : $$\{a+bx+I\mid a,b\in \mathbb Z_5\},$$ I ...
6
votes
2answers
147 views

${\rm Hom}_R(M, R/M) =\{0\} \implies R$ is a field.

Let $R$ be a local ring with maximal ideal $M$. Suppose $M$ is finitely generated. Prove that if ${\rm Hom}_R(M, R/M) =\{0\}$, then $R$ is a field. ${\rm Hom}_R(M, R/M)$ stand for the group of ...
1
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1answer
68 views

how to define size function in Euclidean domain

I was reading about examples of Euclidean domains and their proofs. I encountered one problem on how to define size function for various Euclidean domains. For example in $\mathbb{Z}[i]$ size ...
2
votes
3answers
45 views

$F[x]/\langle x^2\rangle$ is not an integral domain

The ring $F[x]/\langle x^2\rangle$ for an infinite field $F$ is an infinite commutative ring with identity which isn't a domain. I'm still stuck in understanding why is it not a integral domain.i.e. ...
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1answer
173 views

if $R$ is a commutative ring with unity and $A$ is a proper ideal of $R$ ,then $R/A$ is a commutative ring with unity

How should I prove that : if $R$ is a commutative ring with unity and $A$ is a proper ideal of $R$ ,then show that $R/A$ is a commutative ring with unity? commutative part will hold because if $R$ ...
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2answers
113 views

$R/I$ is commutative iff $rs-sr \in I$

How to go with proving this : Let $R$ be a ring and let $I$ be an Ideal of $R$. Prove that the factor ring $R/I$ is commutative iff $rs-sr \in I$ $\forall r$ and $s$ in $R$ any hint how to go with ...
0
votes
1answer
95 views

$\Bbb Z[x]$ is not a principal domain

I know this is already answered here but I am wondering that if the following way to prove is also correct - let $$f(x) = 4x^2+4x+1$$ $$g(x)=4x^2-1$$ Since this is a UFD so a unique gcd will ...
1
vote
1answer
16 views

Why does $\langle 1/p^n+\mathbb{Z}\rangle\supseteq\langle 1/p^m+\mathbb{Z}\rangle$ imply $n\geq m$?

I'm trying to show that if $G=\{a/p^n\in\mathbb{Q}:a\in\mathbb{Z},n\geq0\}$ for a fixed prime $p$, then the quotient $G/\mathbb{Z}$ is Artinian. One minor detail I need to finish is $$ \langle ...
2
votes
1answer
76 views

Is it Possible to Add or Multiply Groups?

I came across a GRE Mathematics Subject test question that said the following: "Find the characteristic of the ring $Z_2 + Z_3$." The explanation of the question starts with the statement that ...
4
votes
1answer
164 views

Morita contexts without tears

My question is: Has anybody seen Morita contexts introduced as it is done below? I first intended this as an answer to the question "Reference request: Morita contexts" by Bey, but then decided to ...
2
votes
1answer
71 views

I understand what a division ring is, but cannot find any examples. Can anyone give me one? [duplicate]

From Wolfram Alpha I have that the definition of a division ring is; "a ring in which every nonzero element has a multiplicative inverse, but multiplication is not necessarily commutative". Can ...
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1answer
61 views

The condition that a ring is a principal ideal domain

If $R$ is a nonzero commutative ring with identity and every submodule of every free $R$-module is free, then $R$ is a principal ideal domain. What I don't know is how to show that every ideal is ...
0
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1answer
138 views

Cardinality of a Quotient Ring

Let $R=\mathbb Z[\xi]$, with $\xi=\frac{1+\sqrt{-19}}{2}$. What is the cardinality of $R/aR$, if $0\neq a\in R$ ? Is the cardinality finite, and equal to the number of cosets ? So if $a$ is fix ...
0
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1answer
53 views

$F$ is a field $\implies$ $1_F \neq 0_F$

I came across a step in a proof which stated : $F$ is a field $\implies$ $1_F \neq 0_F$ . but isn't it in general true even if it isn't a field...
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2answers
83 views

ring homomorphism from $\mathbb Z\oplus\mathbb Z$ into $\mathbb Z\oplus\mathbb Z$.

A question from Gallian which I had in my exam says: Suppose $\phi$ is a ring homomorphism from $\mathbb Z\oplus\mathbb Z$ into $\mathbb Z\oplus\mathbb Z$.What are the possibilities for ...
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2answers
75 views

It's true that a valuation ring $R$ in the quotient field of a normal ring $A$ contain $A$?

Let $A$ be a finitely generated $k$-algebra ($k$ algebraically closed) of dimension one, integrally closed in its quotient field $K$. Let $R\subseteq K$ be a valuation ring. It's true that $A\subseteq ...
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0answers
29 views

Divisibility of a sum in a ring

I can't seem to recall, or find anywhere online an answer to this. If you have a ring $R$, and elements $a,b,c\in R$, what conditions do you need on $R$ to have $a\mid b+c\implies a\mid b$ and $a\mid ...
3
votes
1answer
117 views

Maximal Ideals in $R=\{a+bi:a,b\in \mathbb Z\}$

I've read similar question but please this is not duplicate of Maximal ideals in the ring of Gaussian integers because the answer to it contain PID which I've not yet done etc. $R=\{a+bi:a,b\in ...
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1answer
36 views

the factor ring $\mathbb Z[i]/\langle3-i\rangle$

I can't understand this :if I have : the factor ring $\mathbb Z[i]/\langle3-i\rangle$ and am asked to find elements zero in this ,they are $0,3-i,i(3-i),(3-i)+i(3-i)$. But I can't understand this ...
1
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1answer
42 views

Misunderstanding in Cartan-Eilenberg?

In Cartan Eilenberg's Homological algebra, page 13 it says: If $\Gamma$ is a principal ideal ring, then each ideal $I$ of $\Gamma$ is isomorphic with $\Gamma$, thus $I$ is free and $\Gamma$ is ...
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1answer
64 views

Meaning of $S^{-1}R$ notation

Here are objects defined in an exercise: Let $R$ be a commutative ring. Let $A$ be an ideal of $R$ and $S=\{1+a\mid a\in A\}$. The exercise then makes reference to the prime ideals of $S^{-1}R$. ...
3
votes
1answer
271 views

Ideals, Dedekind domain and $\mathbb{Z}[\sqrt{-3}]$

I have the ideal $\mathfrak{a} = (2, 1 + \sqrt{-3})$ in $\mathbb{Z}[\sqrt{-3}]$. I have to show that $\mathfrak{a} \neq (2)$ but $\mathfrak{a}^{2} = (2)\mathfrak{a}$ and then conclude that ideals do ...
0
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1answer
143 views

Ring homomorphism between real numbers and real valued functions

I was going through chapter 10 in Artin there I found following proposition - Here I am unable to figure out how the homomorphism function will look like in from Real no.'s to Ring of real valued ...
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1answer
18 views

Show that there is no polynomial $f \in (\mathbb{Z}/100\mathbb{Z})$ satisfying f(1)=1 and f(11)=17

As stated in the title. This is part of a homework assignment.
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0answers
87 views

Non-Noetherian subring of F[X,Y]

I am trying to prove that, for a given field $F$, the subring $$R:=\{p(X,Y)=\sum c_{ij}X^iY^j \in F[X,Y] : c_{0j}=c_{j0}=0 \text{ whenever } j>0\}$$ of $F[X,Y]$ is not Noetherian. I think I ...
4
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2answers
81 views

For what natural numbers $n$ is $\mathbf Z/n\mathbf Z$ $[x]/(x^3+x+1)$ a field?

I recently saw this question in the exam of a first abstract algebra course in my college. It shouldn't be too difficult, yet I can't seem to get the solution. Any ideas on how to tackle this?
0
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1answer
53 views

counter example in $R$-modules law

Let $M$ be an $R$-module and $K$, $L$ and $N$ submodules. I would like to find a counterexample to the equality $$N\cap (K+L) =(N\cap K)+(N\cap L)$$ I can prove the equality is true when $K ...
0
votes
3answers
31 views

Quotient Rings of Rings in Several Variables

How should I interpret $$\left(\left(\mathbb{Z[x,y]}/(x) \right)/(y) \right)?$$ Is $$\left(\left(\mathbb{Z[x,y]}/(x) \right)/(y) \right) \cong \left(\left(\mathbb{Z[x,y]}/(y) \right)/(x) \right)?$$ ...
2
votes
2answers
241 views

Prime ideals in $k[x,y]/(xy-1)$.

Let $k$ a field. Let $f$ be the ring injective homomorphism $$ f:k[x] \rightarrow k[x,y]/(xy-1)$$ obtained as the composition of the inclusion $k[x] \subset k[x,y]$ and the natural projection map $ ...
3
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2answers
118 views

group of module homomorphisms is a module

I am trying to solve the following problem: Let $M$ and $N$ be two left $A$-modules. Prove that $Hom_A(M,N)$ has a left $Z(A)$-module structure with: $(a.f)(m)=a.f(m)$. Show $Hom_A(A,N) \cong N$ as ...
0
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2answers
42 views

Set of continuous functions as a ring

In Artin there is a question to test whether the set is a ring or not - $S$ = {Set of all real valued continuous functions} (f+g)(x) = f(x)+g(x) And ...
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1answer
44 views

A monomorphism from a ring to a direct sum

Let $R$ be a ring with a family of ideals $A_i$'s ($i\in I)$. We could consider a well-defined $R$-monomorphism from $R/∩A_i$ to the direct product of $R/A_i$'s sending $r+∩A_i$ to the tuple ...
0
votes
1answer
71 views

On local ring homomorphisms

Suppose I have two local rings $A$ and $B$, and suppose I have $\phi : A \rightarrow B$, which is a ring isomorphism. Does it follow then that $\phi$ is a local ring homomorphism? The point of ...
2
votes
1answer
74 views

What are the units in $R[x_1, x_2, … , x_n]$? Or $R[x]$?

I'm self teaching abstract algebra and I've come across this question...I cannot figure it out! R is not assumed to be a domain so it can't just be the constant polynomials, but whenever I try to ...