#### Date of Award

1988

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

Jurgen Hurrelbrink

#### Abstract

This dissertation examines topics in Algebraic K-Theory, concerning the computation of absolute and relative Milnor groups, K$\sb2$(R) and K$\sb2$(R,I), for both commutative and non-commutative classes of rings, including the relative K$\sb2$ of non-commutative (not necessarily commutative) rings, and the absolute K$\sb2$ of commutative semilocal rings. Our main theorem is a natural extension of a result by Maazen and Stienstra (H. Maazen and J. Steinstra, A presentation of K$\sb2$ of split radical pairs, J. Pure Appl. Algebra 10(1977), 271-294) which determines the relative K$\sb2$ of rings in a commutative setting. We prove the non-commutative analog of this result for local rings. Other results proved in this dissertation include the redundancy of two relations given in Dennis and Stein's presentation for K$\sb2$ of a discrete valuation ring (R. K. Dennis and M. R. Stein, K$\sb2$ of discrete valuation rings, Adv. Math. 18(1975), 182-238) and a proof that a normal form used by Kolster (M. Kolster, K$\sb2$ of Non-Commutative Local Rings, J. Algebra 95(1985), 173-200) does not apply more generally to semilocal rings.

#### Recommended Citation

Russell, Robert B., "On the Relative K(2) of Non Commutative Local Rings." (1988). *LSU Historical Dissertations and Theses*. 4595.

https://repository.lsu.edu/gradschool_disstheses/4595

#### Pages

113

#### DOI

10.31390/gradschool_disstheses.4595