Questions tagged [principal-ideal-domains]

For questions about principal ideal domains: rings without zero divisors where every ideal is principal.

A PID is a type of integral domain where every proper ideal can be generated by a single element. Every Euclidean ring is a PID but the converse is not true. Every PID is a unique factorisation domain but not conversely.

Examples of PIDs are any field, $\mathbb{Z}$, rings of polynomials, Gaussian integers, and Eisenstein integers.

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Trapezoidal Motion Profile Using Discrete Method

I'm trying to program an arduino to generate a Trapezoidal Motion Profile to control a DC motor with a quadrature encoder. Essentially, the user will input the desired Target Position, Max Velocity and Acceleration (decel = -accel) and the code will…
M-R
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Bases for submodules of free modules over a PID

I have proved the following: If $G$ is a free abelian group of rank $n$ and $H$ is a subgroup of $G$, then $H$ is free of rank $m\leq n$. Moreover, there exists a $\mathbb{Z}$-basis $x_1,\ldots,x_n$ for $G$ and $a_1,\ldots,a_m \in \mathbb{Z}$ such…
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Over a PID, $\text{rank}(F/N)=0 \Longleftrightarrow\text{rank}(F)=\text{rank}(N)$?

Let $D$ a PID, $F$ a free module rank $n$, $N$ a submodule of $F$. I want to prove (or find a counterexample) of: $\text{rank}(F/N)=0 \Longleftrightarrow\text{rank}(F)=\text{rank}(N)$ $\text{rank}(F/N)=0\Rightarrow F/N=0 \Rightarrow F=N \Rightarrow…
user203327
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Tensor product of quotient and kernel

In my problem I have a PID $R$, elements $0\neq a,b\in R$ and a map $\phi_a:R\rightarrow R$ where $r\mapsto ar$. Assuming I have done all my previous calculations right I need to prove that \begin{equation} \ker(\phi_a)\otimes_R R/bR =…
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Showing that $1 + \sqrt{5}$ is irreducible in $\mathbb{Z}[\sqrt{5}]$

Consider the ring $\mathbb{Z}[\sqrt{5}]$. How can we show that the element $1 + \sqrt{5}$ is irreducible in this ring?
Ryan
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Looking for help, Aluffi Exercise 5.13, Chapter 6: characterization of PIDs

I quote: "Let $M$ be a finitely generated module over an integral domain $R$. Prove that if $R$ is a PID, then $M$ is torsion-free if and only if it is free. Prove that this property characterizes PIDs. (Cf. Exercise 4.3.)" I am having trouble on…
Doug
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continuous poset w.r.t. Scott topology

I am learning continuous poset by myself. I have conclusion as follows: If $P$ is a continuous poset w.r.t. Scott topology then there is $x\in P$ s.t. for any $y\in P$ and for any open sets $U_x$ and $U_y$ ($x\in U_x$ and $y\in U_y$), $y\in U_x$ or…
flourence
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R is a PID, and a is a nonzero nonunit in R. How can we show R/Ra is an injective module over R?

If we use Baer's criterion then it suffices to show that if there exist a map from an ideal $I$ to $R/Ra$ we must find a map $g$ such that $g\circ i=f$.
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If $N \leq M$, there exist free modules $F_N \leq N$, $F_M \leq M$ with $N = F_N \oplus N_{tor}$, $M = F_M \oplus M_{tor}$ with $F_N \leq F_M$

Let $R$ be a principal ideal domain and $M$ a finitely generated $R$-module. Furthermore, let $N$ be a submodule of $M$. Prove or disprove: there exist free submodules $F_N \leq N$, $F_M \leq M$ with $N = F_N \oplus N_{\mathrm{tor}}$, $M =F_M…
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How $\frac{Z[x,y]}{\langle y+1\rangle}$ is Unique factorization domain?

The question is, Given MCQ, Which of the following is true? (a) $Z[x]$ is principal ideal domain. (b) $Z[x,y]/\langle y+1\rangle$ is a unique factorization domain. (c) If $R$ is a principal ideal domain and $p$ is a non-zero prime ideal, then $R/P$…
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Rank and Nullity is preserved by multiplication of invertible matrices (PID).

I want to show that the rank and nullity of a matrix $A$ whose entries come from a PID are preserved by when $A$ is multiplied by invertible matrices i.e If $A=PBQ$, where $P,Q$ are invertible, rank($A$)=rank($B$), nullity($A$)=nullity($B$). Does…
Jhon Doe
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why we can't write that $\pm2=2r(x) +xs(x)$ instead of $x=2v(x)$?

Prove that the ideal $\langle 2, X\rangle$ in $\mathbb{Z}[X]$ is not principal My attempt : I found the answer here In the proof of the theorem it is written that suppose that the ideal $I=\langle2,X\rangle$ in $\mathbb{Z}[X]$ is principal.Then…
jasmine
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Is $\Bbb Z / n \Bbb Z$ a PID?

I’d like to when $\Bbb Z / n \Bbb Z$ is a PID. I don’t know if depends of the value of $n$, it is true for all $n$ or $\Bbb Z / n \Bbb Z$ is never a PID. No idea. In case it’s true I’d like to see the proof (without group theory) . Thanks!
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In a PID, every irreducible element is a prime element. What's wrong with following?

if $p=ab$ $\implies p|ab$ If $p$ is irreducible then either $a$ or $b$ is a unit. If $a$ is a unit, then $a^{-1}p=b$ or, $b \in

$ $ \implies b=pt \implies p|b$ Thus $p$ is prime Something must be wrong here because I didn't use the fact that…

tatha
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How does an ideal generated by $(x^{2})$ in $Z[x]$?

I was just wondering whether there would exist an element $x^{-2}$ and how would the elements in general look like?
Antimony
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