Additively irreducible sequences in commutative semigroups

Abstract

Let S be a commutative semigroup, and let T be a sequence of terms from the semigroup S. We call T an (additively) irreducible sequence provided that no sum of some of its terms vanishes. Given any element a of S, let Da(S) be the largest length of an irreducible sequence such that the sum of all terms from the sequence is equal to a. In the case that any ascending chain of principal ideals starting from the ideal (a) terminates in S, we find necessary and sufficient conditions for Da(S) to be finite, and in particular, we give sharp lower and upper bounds for Da(S) in case Da(S) is finite. We also apply the result to commutative unitary rings. As a special case, the value of Da(S) is determined when S is the multiplicative semigroup of any finite commutative principal ideal unitary ring.