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On the regularity and defect sequence of monomial and binomial ideals

Keivan BornaAbolfazl Mohajer — 2019

Czechoslovak Mathematical Journal

When S is a polynomial ring or more generally a standard graded algebra over a field K , with homogeneous maximal ideal 𝔪 , it is known that for an ideal I of S , the regularity of powers of I becomes eventually a linear function, i.e., reg ( I m ) = d m + e for m 0 and some integers d , e . This motivates writing reg ( I m ) = d m + e m for every m 0 . The sequence e m , called the of the ideal I , is the subject of much research and its nature is still widely unexplored. We know that e m is eventually constant. In this article, after proving various...

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

Keivan BornaAbolfazl Mohajer-Naser — 2013

Czechoslovak Mathematical Journal

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 p / q I - 1 - K [ x ] , then ( q , f ) 1 ...

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