Fusible rings

Abstract

ABSTRACTA ring R is called left fusible if every nonzero element is the sum of a left zero-divisor and a non-left zero-divisor. It is shown that if R is a left fusible ring and σ is a ring automorphism of R, then R[x;σ] and R[[x;σ]] are left fusible. It is proved that if R is a left fusible ring, then Mn(R) is a left fusible ring. Examples of fusible rings are complemented rings, special almost clean rings, and commutative Jacobson semisimple clean rings. A ring R is called left unit fusible if every nonzero element of R can be written as the sum of a unit and a left zero-divisor in R. Full rings of continuous functions are fusible. It is also shown that if in a ring R, where the ei are orthogonal idempotents and each is left unit fusible, then R is left unit fusible. Finally, we give some properties of fusible rings.