Journal of Algebra, Number Theory: Advances and ApplicationsAuthor's: M. A. Georgiacodis and P. N. Georgiadis
Pages: [29] - [52]
Received Date: March 20, 2015
Submitted by:
DOI: http://dx.doi.org/10.18642/jantaa_7100121471
In Mathematics, it is common practice to identify the carriers of
isomorphic algebraic structures. We believe that this is incompatible
with the rigour and the accuracy which are the main characteristics of
Mathematics.
Specifically, if 
are algebraic structures, A and
E are disjoint sets, B is a proper subset of 
are the restrictions of 
respectively, in 
and f is an isomorphism of
onto 
then it is falsely accepted that 
for every 
and, although A and B are
disjoint sets, A is identified with B and A is
considered as a subset of E.
In this paper we prove that this inaccuracy can be avoided. To do
that, we consider the set 
and using the operations 
and 
we define the operations 
on K in a way that the algebraic
structure 
is isomorphic to 
the operations 
and 
are the restrictions of in 
and literally A is a subset of
K.
Finally, we give two applications of our theorems. In the first
application, we construct the integral domain 
in a way that 
is indeed a proper subset of 
In the second application, we construct the
integral domain 
where 
is the set of polynomials in one
indeterminate X with coefficients in the carrier F of
the field 
in a way that F is a proper subset
of 
algebraic structures, restrictions, isomorphism.
