Displaying similar documents to “Cale Bases in Algebraic Orders”

Uppers to zero in R [ x ] and almost principal ideals

Keivan Borna, Abolfazl Mohajer-Naser (2013)

Czechoslovak Mathematical Journal

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Let R be an integral domain with quotient field K and f ( x ) a polynomial of positive degree in K [ x ] . In this paper we develop a method for studying almost principal uppers to zero ideals. More precisely, we prove that uppers to zero divisorial ideals of the form I = f ( x ) K [ x ] ∩ R [ x ] are almost principal in the following two cases: – J , the ideal generated by the leading coefficients of I , satisfies J - 1 = R . – I - 1 as the R [ x ] -submodule of K ( x ) is of finite type. Furthermore we prove that for I = f ( x ) K [ x ] ∩ R [ x ] we have: – I - 1 ∩ K [ x ] = ( I : K ( x ) I ) . – If there exists...

Star operations in extensions of integral domains

David F. Anderson, Said El Baghdadi, Muhammad Zafrullah (2010)

Actes des rencontres du CIRM

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An extension D ⊆ R of integral domains is t - (resp., t -) if ( I R ) - 1 = ( I - 1 R ) v (resp., ( I R ) v = ( I v R ) v ) for every nonzero finitely generated fractional ideal I of D . We show that strongly t -compatible implies t -compatible and give examples to show that the converse does not hold. We also indicate situations where strong t -compatibility and its variants show up naturally. In addition, we study integral domains D such that D ⊆ R is strongly t -compatible (resp., t -compatible) for every overring R of D . ...

Characterization of irreducible polynomials over a special principal ideal ring

Brahim Boudine (2023)

Mathematica Bohemica

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A commutative ring R with unity is called a special principal ideal ring (SPIR) if it is a non integral principal ideal ring containing only one nonzero prime ideal, its length e is the index of nilpotency of its maximal ideal. In this paper, we show a characterization of irreducible polynomials over a SPIR of length 2 . Then, we give a sufficient condition for a polynomial to be irreducible over a SPIR of any length e .

On the ring of p -integers of a cyclic p -extension over a number field

Humio Ichimura (2005)

Journal de Théorie des Nombres de Bordeaux

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Let p be a prime number. A finite Galois extension N / F of a number field F with group G has a normal p -integral basis ( p -NIB for short) when 𝒪 N ′ is free of rank one over the group ring 𝒪 F ′ [ G ] . Here, 𝒪 F ′ = 𝒪 F [ 1 / p ] is the ring of p -integers of F . Let m = p e be a power of p and N / F a cyclic extension of degree m . When ζ m ∈ F × , we give a necessary and sufficient condition for N / F to have a p -NIB (Theorem 3). When ζ m ∉ F × and p ∤ [ F ( ζ m ) : F ] , we show that N / F has a p -NIB if and only if N ( ζ m ) / F ( ζ m ) has a p -NIB (Theorem 1). When p divides [ F ( ζ m ) : F ] , we show that this...

On wsq-primary ideals

Emel Aslankarayiğit Uğurlu, El Mehdi Bouba, Ünsal Tekir, Suat Koç (2023)

Czechoslovak Mathematical Journal

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We introduce weakly strongly quasi-primary (briefly, wsq-primary) ideals in commutative rings. Let R be a commutative ring with a nonzero identity and Q a proper ideal of R . The proper ideal Q is said to be a weakly strongly quasi-primary ideal if whenever 0 ≠ a b ∈ Q for some a , b ∈ R , then a 2 ∈ Q or b ∈ Q . Many examples and properties of wsq-primary ideals are given. Also, we characterize nonlocal Noetherian von Neumann regular rings, fields, nonlocal rings over which every proper ideal is wsq-primary, and zero...

Making sense of capitulation: reciprocal primes

David Folk (2016)

Acta Arithmetica

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Let ℓ be a rational prime, K be a number field that contains a primitive ℓth root of unity, L an abelian extension of K whose degree over K, [L:K], is divisible by ℓ, a prime ideal of K whose ideal class has order ℓ in the ideal class group of K, and a any generator of the principal ideal ℓ . We will call a prime ideal of K ’reciprocal to ’ if its Frobenius element generates G a l ( K ( a ℓ ) / K ) for every choice of a . We then show that becomes principal in L if and only if every reciprocal prime is not...