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Principal Ideal Ring which is not Integral

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#abstract-algebra #commutative-algebra #principal-ideal-domains
200
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A variation of the argument to prove that $\{m/n:n \text{ is odd },n,m \in \mathbb{Z}\}$ is a PID

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#abstract-algebra #commutative-algebra #rational-numbers #principal-ideal-domains
82
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Why is this element irreducible?

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#abstract-algebra #ring-theory #principal-ideal-domains #unique-factorization-domains
190
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Quotient of normal ring by principal ideal

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#abstract-algebra #ring-theory #commutative-algebra #principal-ideal-domains
198
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$R/(a) \oplus R/(b) \cong R/\gcd(a,b) \oplus R/\operatorname{lcm}(a,b) $

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#abstract-algebra #principal-ideal-domains
2.7k
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Proving a prime ideal is maximal in a PID

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#abstract-algebra #ring-theory #maximal-and-prime-ideals #principal-ideal-domains
502
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Localization of PID, if DVR, is a localization at a prime ideal

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#ring-theory #commutative-algebra #principal-ideal-domains #localization #krull-dimension
334
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Structure theorem for modules implies Smith Normal Form

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#abstract-algebra #ring-theory #modules #principal-ideal-domains
890
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Let R be a PID, B a torsion R module and p a prime in R. Prove that if $pb=0$ for some non zero b in B, then $\text{Ann}(B)$ is a subset of (p)

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#abstract-algebra #ring-theory #modules #principal-ideal-domains
68
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Why can't a finitely generated module over a PID be generated by fewer elements than the number of invariant factors?

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#linear-algebra #abstract-algebra #modules #principal-ideal-domains
129
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Localization of Modules over Dedekind domains

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#abstract-algebra #modules #algebraic-number-theory #principal-ideal-domains #dedekind-domain
55
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Simultaneous bases for free modules over PIDs?

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#abstract-algebra #modules #principal-ideal-domains #free-modules
60
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The ideals of $\mathbb{Z}_{(p)}$ = {$m/n \in \mathbb{Q} : p$ does not divide $n$}.

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#abstract-algebra #ring-theory #ideals #principal-ideal-domains #integral-domain
46
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Show that $M = M_a \oplus M_b$.

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#modules #principal-ideal-domains #direct-sum
46
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Consider the ideal $I = \langle(3,4) \rangle$ of the ring $\mathbb{Z} ×\mathbb{Z}$. Prove that $(\mathbb{Z}×\mathbb{Z})/I$ is not a domain.

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#abstract-algebra #principal-ideal-domains

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