The lattice of ideals of a polynomial semiring (Q1315332)

A semiring \((R,+,\cdot)\) is a nonempty set \(R\) where \((R,+)\) is a commutative monoid with additive identity 0 and the second binary operation satisfies the distributive laws \(a(b + c) = ab + ac\), \((b + c)a = ba + ca\) and \(a0 = 0 = 0a\). By \(R[X]\) we denote the semiring of polynomials over \(R\). Recall that a left ideal \(I\) of \(R[X]\) is said to be subtractive if \(f\in R[X]\) and \(f + g\), \(g\in I\) imply \(f\in I\). The authors prove that for a semiring \(R\), the following statements are equivalent: (1) \(R\) is a ring; (2) every left ideal of \(R[X]\) is subtractive; (3) the lattice of left ideals of \(R[X]\) is modular.