A PROBLEM OF G. Q. WANG ON THE DAVENPORT CONSTANT OF THE MULTIPLICATIVE SEMIGROUP OF QUOTIENT RINGS OF F-2[x]

作者：Zhang, Lizhen; Wang, Haoli^*; Qu, Yongke

来源：Colloquium Mathematicum, 2017, 148(1): 123-130.

DOI：10.4064/cm6707-6-2016

• zero-sum
• Davenport constant
• multiplicative semigroups
• polynomial rings

Given a finite commutative semigroup S (written multiplicatively), denote by D(S) the Davenport constant of S, the least positive integer l such that for any x(1),... x(l) epsilon S there exists a set I not subset of [1,l] for which Pi(i epsilon I) x(i), = Pi(l)(i)=(1) dx(i), the equality being interpreted in the conditional unitization of S to make sense of the left-hand side also in the case when I = empty set and S is not unitary.
Then, let R be the quotient ring of F2[x] by the principal ideal generated by a non-constant polynomial f epsilon F2[x]. Moreover, let S-R be the multiplicative semigroup of the cosets in R, and U(S-R) the group of units of S-R.
We prove that
D(U(S-R) ) <= D(S-R) <= D (U(S-R) ) + delta(f),
where
delta(f) = { 0 if ged (x * (x + 1(F2) ), f) = 1(F2) ,
1 if ged (x * (x + 1(F2) ), f) epsilon {x, x + 1(F2) },
2 if ged (x * (x + 1(F2) ), f) = x * (x + 1(F2) ).
This gives a partial answer to an open problem of G. Q. Wang.

• 出版日期2017
• 单位天津师范大学; 洛阳师范学院; 天津工业大学; 上海大学