Central stable range one for elements and rings

Abstract An element a of a ring R is said to have left central stable range one if whenever $$Ra+Rb=R$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mi>R</mml:mi> <mml:mi>b</mml:mi> <mml:mo>=</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> for any $$b\in R$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> , there exists $$z\in Z(R)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> (the center of R ) such that $$a+zb$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mi>z</mml:mi> <mml:mi>b</mml:mi> </mml:mrow> </mml:math> is a unit of R . A ring R has left central stable range one if all its elements have it. In this work, elements and rings with left central stable range one are studied. The condition is left-right symmetric for such rings.