Nonsimple collections of cells whose inner 2-minors ideals are prime: I

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2026年3月27日

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Lu Zheng,
Jin Guo,
Tongsuo Wu

摘要

In commutative algebra, Qureshi is the first to introduce and study polyominoes and their polyomino ideals. It is known that the polyomino ideal of a simple polyomino is prime. One can then ask for which nonsimple polyominoes their polyomino ideals are prime. In this paper, we extend the work to the collections $\mathcal {P}$ $P$ of cells and introduce a new view, called zero-sum condition, on the labelings on $\mathcal {P}$ $P$ to characterize the elements in the lattice ideal $I_{\Lambda }$ $I_{Λ}$ associated to $\mathcal {P}$ $P$ . This provides us a method to prove the primality of the nonsimple polyominoes constructed by removing an arbitrary hole from a rectangle.