NONTRIVIALITY OF SK1(R[M])

The main result of this paper is as follows: For any commutative regul ar (actually even K,-regular) ring R and any finitely generated interm ediate monoid Z(+)(r):. subset of M subset of Q(+)(r): (for some natur al r) the following conditions are equivalent: (a) M approximate to Z( +)(r), (b) R[M] is K-1-regular, (c) M is seminormal and SK1(R)=SK1(R[M ]) (i.e. the natural homomorphism SK1(R) --> SK1(R[M]) is an isomorphi sm), and, if in addition Omega(R), not equal 0, (d) SK1(R)= SK1(R[M]), where Omega(R), is a module of absolute differentials. The implicatio ns (a) double right arrow (b) double right arrow (c) are well known. I n Sections 8-10 we present examples, further generalizations and appli cations.