On the identification of carriers, II. (Integral domains)
01 Oct 2010-Journal of Interdisciplinary Mathematics (Taylor & Francis Group)-Vol. 13, Iss: 5, pp 509-521
TL;DR: It is proved that when an integral domain is embedded in the integral domain, that is when is isomorphic to a subdomain of, it can be constructed an integraldomain () isomorph to such that is indeed, a sub domain of () .
Abstract: In mathematics, the identification of the carriers of isomorphic algebraic structures, although mathematically inappropriate, is a standard practice. In this paper we deal with the case of the identification of the carriers of integral domains, and we prove that when an integral domain is embedded in the integral domain , that is when is isomorphic to a subdomain of , we can construct an integral domain () isomorphic to such that is indeed, a subdomain of () .
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01 Jan 2015
TL;DR: In Mathematics, it is common practice to identify the carriers of isomorphic algebraic structures as mentioned in this paper, which is incompatible with the rigour and the accuracy which are the main characteristics of Mathematics.
Abstract: 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 ( )( )( )
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TL;DR: In this article, it was shown that the abuse of subfields of a subfield of a field is a mathematical abuse and that this abuse can be avoided by using operations + and · in such a way that the algebraic structure (K, +,.) is a field-isomorphic to while the field is indeed a sub-field of (K +,.).
Abstract: Let be a subfield of the field . If the field is isomorphic to then the sets F 1 and E are usually identified and the field is considered as a subfield of . This is a mathematical abuse and according to R. Godement “Strictly speaking this is false as anything could be” [1]. In this paper we show that this abuse can be avoided. The way to do this is as follows. We consider the set K = (F2 − E) ∪ F1 and, using the operations and , we define on the set K two operations + and · in such a way that the algebraic structure (K, +,.) is a field isomorphic to while the field is indeed a subfield of (K, +,.).
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