0
votes
0answers
44 views

ideals in rings

Let $M$ be the ideal generated by $x^2+2$ in the ring $\ \Bbb Z/3\ \ [x]$. What are the distinct congruence classes in $\ \Bbb Z/3\ \ [x]/I$, is I a prime ideal,a maximal ideal, and is $\ \Bbb Z/3\ \ ...
3
votes
3answers
54 views

Finding indempotents in a quotient ring

I am trying to find the nontrivial indempotents in the ring $\mathbb{Z_3}[x]/(x^2+x+1)$. We can clearly see that $0,1$ are indempotents. I want to prove they are the only ones. Thus I am ...
0
votes
1answer
42 views

Direct Product of Rings and Atoms

Which elements in a direct product $\prod _{\lambda \in \Lambda} R_\lambda$ of rings are atoms? Prove your answer. (Assume this is a commutative ring) My Answer: Atoms of $\prod R_\lambda$ are ...
1
vote
2answers
24 views

If $R$ is $\text{PID}$ and $x \in R$ is irreducible, then $R/(x^k)$ is a local ring.

Suppose $R$ is $\text{PID}$ and let $x \in R$ be irreducible. Let $k \in \mathbb{Z}_{>0}. $Could anyone advise me on how to prove $R/(x^k)$ has a unique maximal ideal? Hints will suffice, thank ...
1
vote
2answers
31 views

Is $2\mathbb{Z}_{12}$ maximal ideal of $\mathbb{Z}_{12} \ ?$

I came across this solution to the a/m problem here: https://sg.answers.yahoo.com/question/index?qid=20110515182700AAsSjeA. The author of the solution claims that $\mathbb{Z}_{12}/2\mathbb{Z}_{12} ...
0
votes
1answer
16 views

An attempt to verify if $\mathbb{Z}_{7^5}$ a local ring with unique maximal ideal $(7) \ .$

A commutative ring $R$ with identity is called a local ring if there exists unique maximal ideal in $R.$ Hence, is $\mathbb{Z}_{7^5}$ a local ring with unique maximal ideal $(7) \ ?$ Here is my ...
0
votes
2answers
20 views

Suppose $R$ is a ring and $\exists n \in \mathbb{Z}_{> 0}$ such that $(ab)^n=ab, \forall a,b \in R.$ Then $ab = 0 $ iff $ba=0 \ ?$

Suppose $R$ is a ring and there exists $n \in \mathbb{Z}_{> 0}$ such that $(ab)^n=ab, \forall a,b \in R.$ Could anyone advise me on how to prove/disprove that $ab = 0$ iff $ba=0, \forall a,b \in R ...
1
vote
1answer
49 views

Nilpotent Elements and Intersection of Prime Ideals

Prove that the set of nilpotent elements of a ring is the intersection of its prime ideals. I know these two useful facts: {nilpotent elements}$=\sqrt{0}$ $\sqrt{I}= \bigcap$ of prime ideals ...
1
vote
0answers
34 views

Module is free of finite rank $\implies$ submodule is free of finite rank?

Let $M$ be $R$-module, where $R$ is commutative ring with $1,$ and $N$ be submodule of $M.$ If $M$ is free of finite rank, so is $N \ ?$ Answer: False. Let $R=M=\mathbb{Z}_{6}$ and ...
1
vote
1answer
23 views

Polynomial ring problem

May I verify if my proof to this problem is correct? Let $p \in \mathbb{P}.$ For $x \in \mathbb{Z},$ let $\overline{x}$ be remainder of $x$ when divided by $p.$ Let $f(X)= \sum^{n}_{i=0}a_iX^i ...
1
vote
1answer
36 views

A problem on non-commutative ring

Let $R$ be a non-commutative ring with $1$ and $a,b\in R$ such that $ab=1 \neq ba.$ Could anyone advise me on how to show there exists $c\in R-\{b\}$ such that $ac=1 \ ?$ Hints will suffice. Thank ...
4
votes
1answer
31 views

If $M$ is Noetherian, then $R/\text{Ann}(M)$ is Noetherian, where $M$ is $R$-module

Let $M$ be a $R$ -module and $\text{Ann}(M)=\{r \in R: rm =0 , \forall m \in M\} .$ Suppose $M$ is Noetherian, could anyone advise me on how to prove $R/\text{Ann}(M)$ is also Noetherian? Hints ...
1
vote
2answers
35 views

Not all ideals are finitely generated

Let $R=\{a_0+a_1X+...+a_nX^n \ | \ a_1,...,a_n \in \mathbb{Q}, a_o \in \mathbb{Z}, n\in \mathbb{Z}_{\geq 0} \}$ and $I=\{a_1X+...+a_nX^n \ | \ a_1,...,a_n \in \mathbb{Q}, n\in \mathbb{Z}^{+} \}.$ ...
1
vote
2answers
26 views

A question on Noetherian $R$ -module. [duplicate]

Let $M$ be Noetherian $R$-module(where $R$ contains $1$) and $\phi:M \to M$ be $R$ -module homomorphism . Suppose $\phi$ is surjective, how do I show that $\phi$ is injective ? Hints will suffice, ...
0
votes
0answers
26 views

Linearly Independent and Span Proof

Let R be a field, M be an R-module, $X \subseteq M$. Show that $X$ is linearly independent if and only if $x$ not $\in$ span (X\ {x}) for each $x \in X$ (I'm not sure how to write the symbol for ...
0
votes
1answer
36 views

Span and Smallest Submodule Proof

Let R be a ring, M a R-module, and $X \subseteq M$ Show that span$(X)$ is the smallest submodule of R containing X. My ideas: Every submodule is contained in its span so $X \subseteq$ span$(X)$ and ...
1
vote
2answers
39 views

Ring with special rules for add and mult

$R$ is a ring of integers with special rules for multiplication and addition. Suppose that $f: \mathbb Z \to R$ is an isomorphism that is defined by $f(x) = x-2$. What integer in the ring $R$ is ...
1
vote
1answer
32 views

If $M,N$ are $R$-modules, then every submodule of $M \times N$ is the form of $U\times V,$ where $U,V$ are submodules of $M,N \ ?$

Let $M,N$ be $R$-modules and suppose $U$ and $V$ are submodules of $M$ and $N$ respectively. I have shown that $U \times V$ is submodule of $M \times N.$ May I know how do I prove/disprove that the ...
-2
votes
1answer
31 views

Let $R$ be a field. What are the $R$-submodules of $R \times R \ ?$

We know that $R$-submodule of $R$ are left ideals of $R.$ Is it also true that $R$-submodule of $R \times R$ are left ideals of $R \times R\ ?$ Please advise on the correct approach to this ...
0
votes
2answers
51 views

Properties of $R/I$

Let $R$ be an integral domain, and let $a$ be an irreducible element of $R$. Let $I$ be the ideal of $R$ generated by $a$. 1.If $R$ is a principal ideal domain, $R/I$ is a field ? True. Since $a$ ...
0
votes
1answer
54 views

Is the property of Euclidean domain inherited via surjective ring homomorphism? [duplicate]

Let $f:R \to S$ be surjective ring homomorphism and $R,S$ be integral domains. Could anyone advise me on how to prove/disprove this statement: If $R$ is Euclidean domain, then $S$ is Euclidean domain. ...
1
vote
1answer
65 views

Ascending chain condition and ring homomorphism

Let $f : R \to S$ be a surjective ring homomorphism between two integral domains. Could anyone advise me on how to prove/disprove the following statements: If $R$ satisfies the ascending chain ...
2
votes
2answers
36 views

Let $R$ be integral domain and $r \not | a.$ If $r$ is prime and $r^k|ab,$ then $r^k|b ?$

Let $R$ be an integral domain and $a,b,r \in R.$ Let $r$ be prime. Suppose there exists positive integer $k$ such that $r^k$ divides $ab$ and $r$ does not divide $a.$ Could anyone advise me on ...
0
votes
0answers
29 views

GCD and LCM Property

Let D = $\mathbb R + X\mathbb C[X]$ Show that $GCD(X, iX) = \mathbb R^\times$ and $LCM(X, iX) = \emptyset$ I have an outline of what to do but don't exactly know who to show all of it... First, ...
0
votes
0answers
11 views

Domain GCD Property

Let D be a domain and $\emptyset \subset A \subseteq D^*$ $d \in GCD(A)$ if and only if (d) is a minimum among the principal ideals containing (A) If $d \in GCD(A)$ then d|a for all $a \in A$ and ...
0
votes
1answer
22 views

Property of GCD in ring

Let D be a domain and $\emptyset \subset A \subseteq D^*$ Show that CD(A)={$d\in D$ | $(A)\subseteq (d)$} I know that I'll need to show both containments to show that the two statements are ...
0
votes
1answer
50 views

GCD Domain Proof

Let $D = \mathbb{R} + X \mathbb{C}[X]$ Show that $\gcd_D(X^2,iX^2)=\emptyset $ Here is my plan so far... (and my questions) Suppose $f \in \gcd_D(X^2,iX^2) $. How do I show that because X is ...
2
votes
2answers
56 views

Irreducibililty of $X^6+X^3+1$ in $\mathbb{Q}[X] \ $

Could anyone advise me on how to prove $X^6+X^3+1$ is irreducible in $\mathbb{Q}[X] \ ?$ I'm thinking of substituting $X=Y+1$ into the equation, do some tedious computations to simplify and use ...
0
votes
0answers
21 views

Operations in a polynomial ring over $\mathbb{F}_5$

Let $f(x)=3x^2 + 4x + 2$ and let $g(x) = 2x + 3$. Perform the following operations in $\mathbb{Z}/5\mathbb{Z}[x]$. (a) $f(x) + g(x)$ (b) $f(x)g(x)$ (c) divide $f(x)$ by $g(x)$. What is the ...
0
votes
2answers
69 views

Irreducible elements and Associates

Show that, in a domain, every associate of an atom is an atom. An atom is the same thing as an irreducible element. I think these two facts will be important to prove this statement: A nonunit ...
0
votes
1answer
43 views

$X^4-5X^2+X+1$ is irreducible in $\mathbb{Q}[X]$

Could anyone advise me on how to efficiently prove $X^4-5X^2+X+1$ is irreducible in $\mathbb{Q}[X] \ ?$ Hints will suffice. Thank you.
1
vote
1answer
34 views

Roots of polynomial in $R[X],$ where $R$ is $\text{UFD}.$

Let $R$ be a $\text{UFD},$ with field of fractions $F$ and let $f(X)=a_0+a_1X+...+a_nX^n \in R[X]$ such that $a_n \neq 0.$ Let $x\in F$ be a root of $f(X).$ Could anyone advise me on how to show ...
0
votes
2answers
40 views

Is this set a subring of $\mathbb{Z}\times\mathbb{Z}$?

Is the set $S = \{(x,-x) : x \text{ is an integer}\}$ a subring of $\mathbb{Z}\times\mathbb{Z}$? I am not sure where to start here. Is $\mathbb{Z}\times\mathbb{Z}$ a matrix? It doesn't seem ...
3
votes
1answer
74 views

Rings (integral domain and fields)

True or false: (1) Every integral domain is a field (2) every field is an integral domain (3) the ring $\mathbb Z$ is a field. (4) the ring $\mathbb Z/(17)$ is a field. (5)The set $\{[0], [2], ...
0
votes
2answers
58 views

Let $R$ be a Noetherian ring. Then all finitely generated $R$-modules are Noetherian

Here is an excerpt of my lecture notes: " Claim I: Let $M$ be $R$- module and $N$ be submodule of $M.$ Then $M$ is Noetherian iff $N, \ M/N$ are Noetherian. Def: The ring $R$ is Noetherian iff the ...
0
votes
1answer
26 views

A problem on $\text{ACCP}$

Let $R$ be a commutative ring. Could anyone advise me on how to prove $R$ has $\text{ACCP}$ (Ascending chain condition for principal ideals) iff every collection of principal ideals of $R$ has maximal ...
0
votes
1answer
22 views

If $R$ is $\text{UFD},$ then $R[X,Y]$ is $\text{UFD}.$

Let $R$ be commutative ring with $1.$ Suppose $R$ is $\text{UFD}.$ Could anyone advise me on how to prove $R[X,Y]$ is $\text{UFD}\ ?$ Thank you.
1
vote
1answer
56 views

Properties of GCD in rings

Let $R$ be subring of integral domain $S.$ Suppose $R$ is $\text{PID}.$ Let $a\in R$ be a greatest common divisor of $r_1,r_2$ in $R$. ($r_1,r_2 \in R$, not both zero). Could anyone advise me on how ...
1
vote
2answers
33 views

Associates in Domains

Let D be a domain and $a, b \in D^*$. Show that $a$ is a proper divisor of $b$ if and only if $b=ax$ for some nonzero nonunit $x$. I'm just really not sure how to start this. Any advice would be ...
0
votes
0answers
17 views

Ring Theory Domain Proof

Let D be a domain. Show that $D[X]^x$=$D^x$. Because D is a domain it means that it is cancellative and D has no nonzero zero divisors. The only units in $D[X]^x$ are the units in $D^x$ so it's ...
1
vote
0answers
62 views

Noetherian Ring and Homomorphic Image

Prove that, if $R$ is Noetherian, then so is each homomorphic image of $R$. I know that by the Fundamental Homomorphism Theorem this is the same as showing that if $R$ is Noetherian, then so is ...
0
votes
0answers
28 views

Ring Embeds in Monoid Ring

Let $(S,+)$ be a nontrivial commutative monoid and $R$ be a ring. Prove that $R$ embeds in $R[X;S]$ via $a \to aX^0$ I'm not exactly sure how to approach this... I think I may need to use the fact ...
0
votes
1answer
28 views

Greatest common divisors in Integral Domain

Let $R$ be an integral domain and $r,s\in R-\{0\}$ such that $\text{gcd}(r,s)=g.$ Suppose $\text{gcd}(kr,ks)$ exists, where $k \in R -\{0\}.$ Could anyone advise me on how to prove $kg= ...
1
vote
2answers
32 views

A problem on $\text{UFD}$

Let $R$ be a $\text{UFD}$, and let $a,b,c \in R$ such that $1=\text{gcd}(a,b).$ Suppose $a |c, \ b|c.$ Could anyone advise me on how to prove $ab |c \ ?$ How do I use the fact that every nonzero non ...
1
vote
0answers
40 views

Integral Domain and PID Proof

Prove that, in a domain, $(a)=(b)$ iff $a = bu$ for some unit $u$. By $(a)=(b)$, it also means that $a\mid b$ and $b\mid a$ so we can write them as $a=bu$ and $b=av$ for some $u, v \in R$ where $u$ ...
2
votes
1answer
38 views

Divisibility and Principal Ideal Domain Proof

Let R be a ring. Show that a|b iff $b \in (a)$ iff $(b) \subseteq $ (a). I first just want to write out what I know about this statement: a|b means that a divides b or a is divisible by b and there ...
0
votes
3answers
95 views

Maximal Proper Ideal is a field proof

Show that a proper ideal M of a commutative ring R is maximal if and only if R/M is a field. What I know: Because M is a proper ideal $M \neq R$. The ideal M is maximal if it is a maximal element ...
2
votes
1answer
55 views

A question on rings

Let $R$ be an integral domain and $S$ be subring of $R$ with $1_R=1_S.$ Let $T=\{f(x) \in R[X]: f(0) \in S\}.$ Suppose $R[X]$ satisfies ascending chain condition for principal ideals, $ACCP.$ Could ...
0
votes
2answers
101 views

How to show that $x$ becomes a root of $p(x)$ in $F[x]/(p(x))$

$F$ is a field, $p(x)$ is irreducible polynomial at $F[X]$. $K=F[X]/\left<p(x)\right>$. For every $a\in F$ we will mark: $\bar{a}=\left<p(x)\right>+a$. Now, the question is: How do I show ...
0
votes
2answers
39 views

Prove Well-defined functions

This problem is from my textbook, chapter of Isomorphism and Invariant. Which of the following functions are well-defined? Prove your answers. (a) $f:\mathbb{Q}\to \mathbb{Q}$ defined by $f(\frac a ...