Row reduced matrices and annihilator semigroups 1

Throughout we are discussing matrices with entries from a field K. It was first proved in 1] that a product of row reduced matrices is row reduced. This means that the set of row reduced matrices in any matrix ring form a semigroup. It is also the case that every matrix A ? M_n(K)has the property that it has the same right annihilator as its row reduced form, and distinct row reduced matrice have distinct right annihilators. Let R be a ring. Motivated by these observations, we call a multiplicative semigroup S in R a right annihilator semigroup for R if every element in R has the same right annihilator as exactly one element in S. Reasoning that row reduced matrices are very important we study semigroups that share their formal properties. Ultimately we would like to know all right annihilator semigroups in M_n(K).This seems to be a formidable task. Here we determine all right annihila-tor semigroups in M₃(K) up to a change of basis, that is conjugation.