On The K₂ Of A Ring With Stable Range One

This dissertation is focused on the K2 of a ring with stable range 1.In Chapter 2 we give a description of the K1 of a special class of rings withstable range 1, called the semiperfect ring, in terms of its particular structure.This is a generalization of Guo's result about the K1 of a semiperfect ring undera certain condition.In Chapter 3 we investigate two particular types of diagonal elements in theSteinberg group over an arbitrary ring, called the H-type element and the W-type element, respectively, and give their basic properties, such as the conjugationformula, the inverse formula and the shifting formula, etc. The two types ofelements can be viewed as n-times counterparts of the classical Steinberg symbolsand Dennis-Stein symbols.Chapter 4 starts with the normal form of the Steinberg group over a ringwith stable range 1, from which it follows that the K2 of a ring with stable range1 is contained in the subgroup of the Steinberg group generated by all the three-times W-type elements. Then we discuss the left and right multiplications onthe normal form of the Steinberg group by the element w12(1), which leads to animportant relation of the three-times H-type elements. In the end, we make anattempt to generalize Keune's"cyclic formula"of three-times W-type elements inthe Steinberg group over a commutative ring.