1
vote
2answers
30 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
21 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
25 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
26 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
33 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
24 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
29 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
42 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
47 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
44 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
30 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
20 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
10 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
16 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
37 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
54 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
20 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
34 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
42 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
31 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
39 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 ...
-1
votes
0answers
20 views

Definition of Subrings

(5)The set $\{[0], [2], [4]\}$ is a subring of $\mathbb Z(6)$. I bieleve this is false. It is closed under add/mult but does it have a 0 or an x where a + x = 0 is satisfied in S? What does it mean ...
3
votes
1answer
68 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
30 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
23 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
20 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
31 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
33 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
27 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
27 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
31 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
31 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
33 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
57 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
53 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
98 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
30 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 ...
0
votes
0answers
59 views

Show that Weyl algebra is noetherian

Let $k$ be a field. I want to show that the ring $D=k\left[x_1,x_2,\dots,x_n,\frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2},\dots,\frac{\partial}{\partial x_n}\right]$ which acts on ...
-2
votes
1answer
82 views

Help on abstract algebra proof?

Similar question here Let $R$ be the set of all integers with alternative ring operations defined below. Show that $\Bbb Z$ is isomorphic to $R$. The difference is that in attempting to answer my own ...
-4
votes
2answers
92 views

Help with Theorem III.3.11 in Hungerford's algebra book

I need help to prove part (i) of this theorem which I couldn't prove. Any help would be appreciated. Thanks in advance.
3
votes
1answer
81 views

$\mathbb{Z}/m\mathbb{Z}$: A Complete Set of Representatives

So, I'm letting ${\scr{A}}=\{a_1,\dots,a_n\}$ be a complete set of representatives (C.S.R.) for $\mathbb{Z}/m\mathbb{Z}$. I'm considering all $b\in \mathbb{Z}$ and seeing if ${\scr{B}}=\{a_1+b, a_2+b, ...
0
votes
1answer
63 views

$M\cong_{R\operatorname{-Mod}}N$ if and only if $M\cong_{\operatorname{Ab}}N$ (Warning: This statement is FALSE!)

I was rather surprised by the fact that two modules are isomorphic if and only if their abelian group structures are isomorphic. I might just sketch the proof here. Given a ring homomorhpism ...
2
votes
2answers
66 views

Tests/ invariants for module isomorphisms

It two modules are indeed isomorphic, then it is often not too difficult to find an isomorphism since most of the time it is just the natural map. However, it takes some time for me to prove that two ...
1
vote
1answer
274 views

Does every noninvertible element of a commutative ring lie in a proper maximal ideal?

More formally stated: Prove that if $R$ is a commutative ring with $1$, then every element of $R$ that is not invertible is contained in a proper maximal ideal. I know I have to assume Zorn's Lemma, ...
3
votes
2answers
114 views

In general how to prove or disprove certain types of ideal?

i've come across a lot of questions recently that ask you whether or not there exist certain kinds of ideal, say; does there exist an ideal$ J $of $\mathbb{Z}[i]$ for which $\mathbb{Z}[i] /J$ is a ...
1
vote
0answers
75 views

proof that $P^{(n)}$ are primary when $P$ is prime

I am looking for an alternate proof of the fact that in a commutative ring $R$ with a prime ideal $P$, the ideal $P^{(n)}=P^n R_P\cap R$ is primary. I understand once we localize, $P^n R_P$ is a power ...
5
votes
3answers
164 views

Proving that $\mathbb C[C_7]\cong\bigoplus_{i=1}^7\mathbb C$

Let $R$ be the group ring $\mathbb C[C_7],$ where $C_7=\{1,g,\ldots,g^6\}$ is a cyclic group. I would like to prove that $$\mathbb C[C_7]\cong\bigoplus_{i=1}^7\mathbb C.$$ I was thinking that I ...