ON COMPLETABLE UNIMODULAR ROWS OVER SEMIRINGS

Abstract

In Ring theory, the unimodular rows play an important role in studying structures of Hermite rings and other important classes of rings. The basic calculus of unimodular rows was completely described by T.Y. Lam, P.M. Cohn,… especially, completable unimodular rows. According to T. Y. Lam (1978), a ring is right Hermite if any finitely generated stably free right module over the ring is free, and this is equivalent to requiring that any unimodular row on ring can be completed to a invertible matrix. However, when computing the completable unimodular rows on semirings, some properties are no longer true as in rings, and now there are not many research results about this problems. In this paper, we prove some basic properties of unimodular rows over abitrary semirings; indicate a class of semirings in which set of unimodular rows and set of completable unimodular rows are not same; prove the necessary and sufficient conditions for all unimodular rows on commutative semirings can be completed to invertible matrices; describe structure of completable unimodular rows on class of zerosumfree semirings satisfying some given conditions.