On The Characterization Of The Unit Groups Of A Class Of Total Quotient Rings

ABSTRACT

Let R be a completely primary nite ring of characteristic pk, where p is a prime and k is

a positive integer. Such nite rings have been studied extensively in recent years and the

tools necessary for describing completely primary nite rings have been available for some

time. However, their characterization is not exhaustive. Several attempts have been made

in the recent past in the characterization of the unit groups of this class of rings though not

in general. The characterization of a nite abelian group is precisely known and from the

fundamental theorem of nitely generated abelian groups, it has been represented as a direct

product of cyclic groups. If R is a nite eld then U(R) the unit group of R is cyclic. Suppose

S is a saturated multiplicative subset of R so that RS is a total quotient ring of R obtained

by the localization of R at the maximal ideal J(R): Since R is local, then R =

RS: We have

characterized the unit groups of RS; denoted U(RS) as U(RS) = Zpr􀀀1

􀀀1

1 + J(RS)

by

determining the generators of the group 1

1 + J(RS): The methods used include construction

of RS using the method of idealization, proofs and verication of the proposed statements

and claims. The results obtained, provide an alternative understanding of the structures of

unit groups of these classes of nite rings and provides a partial solution to the problem of

isomorphic rings with similar groups of units. This results also nd practical applications in

various elds including Computer Algebra and forms and structure of elements in chemistry.