# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $I=\{a_1X+...+a_nX^n \ | \ a_1,...,a_n \in \mathbb{Q}, n\in \mathbb{Z}^{+} \}.$ ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $a$ ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $X^4-5X^2+X+1$ 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $u$ ...
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### 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 ...
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### 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 ...
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### 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 ...
### 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 ...
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 ...