We introduce the notion of numerical semigroups generated by concatenation of arithmetic sequences and show that this class of numerical semigroups exhibit multiple interesting behaviors.

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In this paper we study primality and primary decomposition of certain ideals which are generated by homogeneous degree 2 polynomials and occur naturally from determinantal conditions. Normality is derived from these results.2010 Mathematics Subject Classification. Primary 13C40, 13P10.

Bresinsky defined a class of monomial curves in [Formula: see text] with the property that the minimal number of generators or the first Betti number of the defining ideal is unbounded above. We prove that the same behavior of unboundedness is true for all the Betti numbers and construct an explicit minimal free resolution for the defining ideal of this class of curves.

In this paper we compute Gröbner bases for determinantal ideals of the form I 1 (XY ), where X and Y are both matrices whose entries are indeterminates over a field K. We use the Gröbner basis structure to determine Betti numbers for such ideals.2010 Mathematics Subject Classification. Primary 13P10; Secondary 13C40, 13D02.