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Figure from:Standard pairs for monomial ideals in semigroup rings

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FIGURE 1. Examples of ideals in two-dimensional affine semigroup rings

Figure 1 Examples of ideals in two-dimensional affine semigroup rings

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To complete the proof, we show that, if the overlap class of (a, F’) is not maximal with respect to divisibility and b € a + NF, then (J : t®) is not prime. Since [a, F] is not maximal, there is a standard pair (a’, F’) of J, whose overlap class is maximal with respect to divisibility, and such that (a, F’) divides (a’, F’). In particular, there is c € NA such that b+ c € a’ + NF. Note that c ¢é NF as (a, F’) and (a’F’) are not in the same overlap class. Since (a’, F’) is a standard pair of J, follows that t° ¢ (I : t°). By the previous argument, however, since c € NA \ NF andb+c€a'+NF, we have ¢° € (I : °°), which implies t?° € (J : t?). We conclude that (J : t) is not prime.
To complete the proof, we show that, if the overlap class of (a, F’) is not maximal with respect to divisibility and b € a + NF, then (J : t®) is not prime. Since [a, F] is not maximal, there is a standard pair (a’, F’) of J, whose overlap class is maximal with respect to divisibility, and such that (a, F’) divides (a’, F’). In particular, there is c € NA such that b+ c € a’ + NF. Note that c ¢é NF as (a, F’) and (a’F’) are not in the same overlap class. Since (a’, F’) is a standard pair of J, follows that t° ¢ (I : t°). By the previous argument, however, since c € NA \ NF andb+c€a'+NF, we have ¢° € (I : °°), which implies t?° € (J : t?). We conclude that (J : t) is not prime.
FIGURE 5. A primary decomposition of (x”y”, xy) in k[x, ry, ry?
FIGURE 5. A primary decomposition of (x”y”, xy) in k[x, ry, ry?
3.3. Irreducible Decomposition. We now address the irreducible decomposition of monomial ideals in semigroup rings using standard pairs. While the existence of monomial irreducible decomposition of monomial ideals in semigroup rings is known [22, Corollary 11.5, Proposi- tion 11.41], an effective combinatorial description of such a decomposition was missing from the literature before this work. As a side note, we recall that monomial ideals in semigroup rings can be viewed as binomial ideals in polynomial rings, and mention that binomial ideals do not in general have irreducible decompositions into binomial ideals [19].
3.3. Irreducible Decomposition. We now address the irreducible decomposition of monomial ideals in semigroup rings using standard pairs. While the existence of monomial irreducible decomposition of monomial ideals in semigroup rings is known [22, Corollary 11.5, Proposi- tion 11.41], an effective combinatorial description of such a decomposition was missing from the literature before this work. As a side note, we recall that monomial ideals in semigroup rings can be viewed as binomial ideals in polynomial rings, and mention that binomial ideals do not in general have irreducible decompositions into binomial ideals [19].
FIGURE 8. An irredundant irreducible decomposition of J = J, Jz in k{[x?, y, ry].
FIGURE 8. An irredundant irreducible decomposition of J = J, Jz in k{[x?, y, ry].