The coordinate ring of a simple polyomino

Abstract

In this paper it is shown that a polyomino is balanced if and only if it is simple. As a consequence one obtains that the coordinate ring of a simple polyomino is a normal Cohen-Macaulay domain.2010 Mathematics Subject Classification. 05B50, 05E40, 13G05.

Nonsimple polyominoes and prime ideals

“…Proof. We may assume that B(P) ∩ B(P I ) = ∅; otherwise, P is a simple polyomino (see Figure 2) and, as was stated, the result follows from [4] and [8]. Here (S/I P c ) xc is the localization of (S/I P c ) xc at x c .…”

“…Thus, in order to prove that S/I P c is an integral domain, it suffices to show that (S/I P c ) xc = S xc /(I P c ) xc is an integral domain. For this, we will show that (I P c ) xc = I P ′ , where P ′ is a simple subpolyomino of P c , which guarantees that (I P c ) xc is a prime ideal ( [4] and [8]).…”

“…As was stated in Introduction, one of the most exciting algebraic problems on polyominoes is when a polyomino ideal is a prime ideal. The fact ( [4] and [8]) that the polyomino ideals of simple polyominoes are prime seems to be of interest. However, it turns out that these binomial ideals belong to a subclass of binomial ideals arising from Koszul bipartite graphs ( [7]).…”

“…Such a binomial f a,b is said to be an inner 2-minor of P. Write I P for the ideal generated by all inner 2-minors of P. Especially, when P is a polyomino, we say that I P is the polyomino ideal of P. Now, one of the most exciting algebraic problems on polyominoes is when a polyomino ideal is a prime ideal. It is known ( [4] and [8]) that if a polyomino P is simple, then its polyomino ideal I P is a prime ideal. The polyomino ideals arising from simple polyominoes, however, turn out to be well-known ideals [7] arising from Koszul bipartite graphs.…”

Toric representations of algebras defined by certain nonsimple polyominoes

“…Let P be a polyomino and K be a field. We denote by I P , the polyomino ideal attached to P, in a suitable polynomial ring over K. It is natural to investigate the algebraic properties of I P depending on shape of P. The classes of polyominoes whose polyomino ideal is prime have been discussed in many papers, including [3,2,4,6] .…”