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) = Zpr1
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.
Wekesa, A (2021). On The Characterization Of The Unit Groups Of A Class Of Total Quotient Rings. Afribary. Retrieved from https://afribary.com/works/on-the-characterization-of-the-unit-groups-of-a-class-of-total-quotient-rings
Wekesa, Aldrin "On The Characterization Of The Unit Groups Of A Class Of Total Quotient Rings" Afribary. Afribary, 08 May. 2021, https://afribary.com/works/on-the-characterization-of-the-unit-groups-of-a-class-of-total-quotient-rings. Accessed 25 Nov. 2024.
Wekesa, Aldrin . "On The Characterization Of The Unit Groups Of A Class Of Total Quotient Rings". Afribary, Afribary, 08 May. 2021. Web. 25 Nov. 2024. < https://afribary.com/works/on-the-characterization-of-the-unit-groups-of-a-class-of-total-quotient-rings >.
Wekesa, Aldrin . "On The Characterization Of The Unit Groups Of A Class Of Total Quotient Rings" Afribary (2021). Accessed November 25, 2024. https://afribary.com/works/on-the-characterization-of-the-unit-groups-of-a-class-of-total-quotient-rings