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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.

897 questions
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Show that $R[x,y]$ is not a Principal Ideal Domain

Let $R$ be a commutative ring with $1$. Prove that a polynomial ring in more than one variable over $R$ is not a P.I.D.. In order to show this is not a P.I.D. what do i need to show.
user146269
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Non-integer domain which every ideal is a principal ideal

Let $F$ be field and $A=F[t]\setminus (t^2)$, where $(t^2)$ is the ideal of $F[t]$ (a) Show that every ideal of $A$ is principal ideal (b) Find all prime ideals of $A$ I know $A$ is not integer domain because $t^2$ is reducible, So it is just…
fivestar
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Is quantum torus a principal ideal domain?

For a quantum torus $C_q[x_1^{\pm1}, ...,x_n^{\pm1}]$ satisfying $x_ix_j=q_{ij}x_jx_i$. Question: Is this quantum torus a principal ideal domain?
victor
  • 31
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Is PID cyclic? I cannot understand quotient ring is cyclic

In the textbook, it says “If $R$ is a principal ideal domain and $r$ is an elt of $R$, then the quotient ring $R/(r)$ is a cyclic R-module. I dont know why $R/(r)$ is cyclic. Thank you in advance.
Cjk34
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Find the kernel of a ring homomorphism $f(x)\to f(\sqrt 2)$

Let $$\phi:\mathbb Z[x]\longrightarrow \mathbb R$$, where $$\phi(f(x))=f(\sqrt 2)$$ find $$\operatorname{Ker}(\phi)=\{f(x)\in \mathbb Z[x]\mid f(\sqrt2)=0\}$$ 1st) I wanted to use isomorphism theorem since we have $\mathbb R$-field so…
Micheal Brain Hurts
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Commutative Rings: Principal Ideal Generator Question

Here is a definition: We say that an ideal, $I$, of a commutative ring $(R,+, \cdot) $ is principal iff $I==\lbrace x\cdot r: r\in R\rbrace$. My question is how $=\lbrace x^m :m\in \mathbb{Z}\rbrace$ (where the power just here is addition)…
W. G.
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Principal Ideal Example

I am confused as to how to deal with basic polynomials in the sense of principal ideals. For example, let $I=\lbrace p(x) \in \mathbb{Z}[X] : p(0)=7j \text{ for some } j\in \mathbb{Z} \rbrace$ be an ideal of $\mathbb{Z}[X]$ (the ring of all…
W. G.
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0
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algebraic poset

I learn domain theory and stack in definition of algebraic poset. Recall $P$ is algebraic if for every $x\in P$,the set of compact element $y$ below $x$ is directed and has $x$ as least upper bound. From this definition,I have a question: whether…
flourence
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Unit in Principal Ideal Domain

So I have a statement that say "Let D be a principal ideal domain, if a is unit then (a)=D". How it could be? I don't understand, can anyone explains it to me? Thank you.
Dicky Ardiyantoro
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To check whether given ring is principal ideal domain?

Is $\mathbb{Z}_6[x]$ is principal ideal domain ? no, we know that polynomial ring $\mathbb{F}[x]$ is field iff $\mathbb{F}$ is field here $\mathbb{Z}_6$ is not field. consider ideal $\langle{x,2}\rangle$ which is not principal ideal in…
dipali mali
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ED,PID and UFD and the relation between them

Let R be a Commutative ring with unity, such that R[x] is UFD. If R[x] is a PID then it is a Eucledian Domain? Is the last statement about being eucledian domain correct?
T.Pal
  • 77
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Examples of PID's with finitely many maximal ideals

Do you know some examples of principal ideal domains which have finitely many maximal ideals? More generally, do you know how to build such domains? I don't look for fields and discrete valuation rings. Thank you.
Ángel Valencia
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Is $\mathbb{Q}$ a principal ideal domain?

Is $\mathbb{Q}$ a principal ideal domain? I know that $\mathbb{Z}$ is, but I'm not sure about $\mathbb{Q}$.
user292184