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New posts in principal-ideal-domains
Are all subrings of the rationals Euclidean domains?
number-theory
commutative-algebra
ring-theory
principal-ideal-domains
Ring of Polynomials is a Principal Ideal Ring implies Coefficient Ring is a Field?
abstract-algebra
commutative-algebra
ring-theory
principal-ideal-domains
How to determine which classes of integral domains a quadratic integer ring is in? [duplicate]
ring-theory
algebraic-number-theory
principal-ideal-domains
unique-factorization-domains
Why any field is a principal ideal domain?
abstract-algebra
ring-theory
field-theory
ideals
principal-ideal-domains
Can any set of $n$ relatively prime elements be extended to an invertible matrix?
linear-algebra
principal-ideal-domains
Isomorphism of direct sums $R/a \oplus R/b \cong R/{\rm lcm}(a,b)\oplus R/\gcd(a,b)$
abstract-algebra
principal-ideal-domains
Dedekind domain with a finite number of prime ideals is principal
abstract-algebra
ring-theory
commutative-algebra
principal-ideal-domains
dedekind-domain
Example of a domain where all irreducibles are primes and that is not a GCD domain
ring-theory
divisibility
principal-ideal-domains
unique-factorization-domains
If $A$ is a Principal Ideal Domain, and $\mathfrak{a}$ its ideal. prove that $\frac{A}{\mathfrak{a}}$ is also a Principal Ideal Domain.
ideals
principal-ideal-domains
Show that every ideal of the ring $\mathbb Z$ is principal
abstract-algebra
ring-theory
ideals
principal-ideal-domains
For which $d$ is $\mathbb Z[\sqrt d]$ a principal ideal domain?
abstract-algebra
ring-theory
algebraic-number-theory
principal-ideal-domains
Proving the quotient of a principal ideal domain by a prime ideal is again a principal ideal domain [closed]
abstract-algebra
principal-ideal-domains
A prime ideal of a polynomial ring over a PID can be generated by two elements. [duplicate]
abstract-algebra
maximal-and-prime-ideals
principal-ideal-domains
If any "dividing" chain "terminates" at some point, does that imply an integral domain being P.I.D.?
abstract-algebra
ring-theory
ideals
principal-ideal-domains
A case where a UFD is a PID
abstract-algebra
ring-theory
principal-ideal-domains
unique-factorization-domains
Showing that $x+x^2$ belongs to an ideal in $\mathbb{Z}_2[x]$
abstract-algebra
ring-theory
field-theory
ideals
principal-ideal-domains
Let $D$ be an integral domain and let $c\in D$ be irreducible in $D$. Show the ideal $(x,c)$ in $D[x]$ is not principal. [duplicate]
abstract-algebra
ring-theory
field-theory
principal-ideal-domains
Euclid's Lemma in a PID: irreducibles are prime: $ \pi\mid ab\Rightarrow \pi\mid a\,$ or $\pi\mid b$
abstract-algebra
number-theory
ring-theory
proof-explanation
principal-ideal-domains
Every principal ideal domain satisfies ACCP.
abstract-algebra
ring-theory
principal-ideal-domains
For which values of $d<0$ , is the subring of quadratic integers of $\mathbb Q[\sqrt{d}]$ is a PID?
ring-theory
algebraic-number-theory
principal-ideal-domains
unique-factorization-domains
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