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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Dedekind domain with a finite number of prime ideals is principal

I am reading a proof of this result that uses the Chinese Remainder Theorem on (the finite number of) prime ideals $P_i$. In order to apply CRT we should assume that the prime ideals are coprime, i.e. the ring is equal to $P_h + P_k$ for $h \neq k$,…
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Ring of Polynomials is a Principal Ideal Ring implies Coefficient Ring is a Field?

I read this proof that if $D$ is an integral domain and $D[X]$ is a principal ideal domain, then $D$ is a field. My question is if the requirements can be relaxed a bit, namely: Is it true that if $D$ is a commutative unitary ring and $D[x]$ is a…
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Why any field is a principal ideal domain?

Why any field is a principal ideal domain? According to the definition of P.I.D, first, a ring's ideal can be generated from a single element; second, this ring has no zero-divisor. This two conditions make a ring P.I.D. But how to prove any field…
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Are all subrings of the rationals Euclidean domains?

This is a purely recreational question -- I came up with it when setting an undergraduate example sheet. Let's go with Wikipedia's definition of a Euclidean domain. So an ID $R$ is a Euclidean domain (ED) if there's some…
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Does there exist a ring which is not a principal ideal ring and which has exactly six different ideals?

Does there exist a ring which is not a principal ideal ring and which has exactly six different ideals? (For me a ring is commutative with a unit element.) I can show that any ring having at most five ideals is a principal ideal ring. EDIT: The…
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Prime elements in $\mathbb{Z}[\sqrt{2}]$

What are the prime elements in the ring $\mathbb{Z}[\sqrt{2}]$? Note that since the ring is a PID (and thus a UFD) then prime = irreducible. Even more, it is Euclidean with respect to the absolute value of the norm: $N(a+b\sqrt{2})=a^2-2b^2$ so it…
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Proofs of the structure theorem for finitely generated modules over a PID

I'm looking for different proofs (references or sketch of main ideas) of the structure theorem for finitely generated modules over a PID. If possible, a comparison in terms of clarity, elegance or usefulness would be appreciated.
Weaam
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Euler's remarkable prime-producing polynomial and quadratic UFDs

Good example of a polynomial which produces a finite number of primes is: $$x^{2}+x+41$$ which produces primes for every integer $ 0 \leq x \leq 39$. In a paper H. Stark proves the following result: $X_{n}$ (the ring of "algebraic integers" in…
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Ring of trigonometric functions with real coefficients

Let $R$ be the ring of functions that are polynomials in $\cos t$ and $\sin t$ with real coefficients. Prove that $R$ is isomorphic to $\mathbb R[x,y]/(x^2+y^2-1)$. Prove that $R$ is not a unique factorization domain. Prove that $S=\mathbb…
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Can we characterize all infinite PID s whose group of units is singleton?

I am looking for a way to characterize all infinite PID s having exactly one unit i.e. invertible element ( finite PID s are not interesting , they are all fields ) . The only such example I know of is $\mathbb Z_2[x]$ . Towards characterizing such…
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Let $R$ be a commutative ring. If $R[X]$ is a principal ideal domain, then $R$ is a field.

I've just read a proof of the statement: Let $R$ be a commutative ring. If $R[X]$ is a principal ideal domain, then $R$ is a field. In one part of the proof there is a step which I don't understand. I'll copy the proof: Let $u\in R$ be…
user100106
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Krull dimension of polynomial rings over noetherian rings

I want to prove the following theorem concerning Krull dimension: Theorem If $A$ is a noetherian ring then $$\dim(A[x_1,x_2, \dots , x_n]) = \dim(A) + n$$ where $\dim$ stands for the Krull dimension of the rings. Thus, $\dim(K[x_1,x_2, \dots ,…
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Why is $(2, 1+\sqrt{-5})$ not principal?

Why is $(2, 1+\sqrt{-5})$ not principal in $\mathbb{Z}[\sqrt{-5}]$? Say $(2,1+\sqrt{-5})=(\alpha)$, then since $2\in(2,1+\sqrt{-5})$ we have $2\in (\alpha)$, so $\alpha\mid2$ in $\mathbb Z[\sqrt{-5}]$. Writing $2=\alpha\beta$ in $\mathbb…
inequal
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Ring of formal power series over a principal ideal domain is a unique factorisation domain

An exercise in my algebra course book asks to prove that if $R$ is a PID, then $R[[x]]$ is a UFD, where $R[[x]]$ is the ring of formal power series over $R$. After some failed attempts at proving the ACC I visited Wikipedia, which comments: If $R$…
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Finitely generated modules over PID

Let $A$, $B$, $C$, and $D$ be finitely generated modules over a PID such that $A\oplus B\cong C\oplus D$ and $A\oplus D\cong C\oplus B$. Prove that $A\cong C$ and $B\cong D$. The only tool I have is the theorem about finitely generated modules,…
user6697
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