Sums of units in Baer and exchange rings

In this paper, we prove that every element in an exchange ring R with artinian primitive factors is n-tuplet-good iff 1R is n-tuplet-good. Also, we show that for such rings the full matrix ring Mn(R) (for n ≥ 2) is n-tuplet-good. In [7], Fisher and Snider proved that every element of a strongly π-regular ring R with 1/2 ∈ R is 2-good. We prove the same result under new condition and show that these rings are twin-good. We also consider the conditions under which endomorphism ring of a finitely generated projective module M over unit regular ring L is 2-tuplet-good. The main result of [14] states that regular self-injective rings are n-tuplet-good if such rings has no factor ring isomorphic to a field D with |D| n+2. We generalized this result to regular Baer rings proving that every regular Baer ring R that has no factor ring isomorphic to a field of order less than n + 2, is n-tuplet-good.