NeutroAlgebra of Ideals in a Ring

NeutroAlgebra of Ideals in a Ring

Ilanthenral Kandasamy, Vasantha W. B., Florentin Smarandache
Copyright: © 2022 |Pages: 14
DOI: 10.4018/978-1-6684-3495-6.ch015
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Abstract

In this chapter, the authors define operations ‘+' and ‘×' on the substructure ideals of the ring. They find under what conditions the collection of ideals are NeutroAlgebra under + and ×. The study is carried out in the case of Z and Z_n, the ring of integers and ring of modulo integers, respectively, where n is a composite number. This study is extended to the case of polynomial rings R[x], Q[x], Z[x], and Z_n [x] where R and Q are real and rational, respectively. Some open problems based on this study are proposed.
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Background

Here some of the basic concepts are recalled to make this chapter a self-contained one. Mainly the definition of NeutroAlgebra are recalled and described as this concept is very new.

A NeutroAlgebra is an algebra with at least one Neutro operation or one Neutro axiom (axiom that is true for some elements, indeterminate or false for the other elements) Smarandache (2020) and Smarandache and Hamidi (2020). A partial algebra has at the minimum one partial operation, and all of its axioms are classical. Smarandache and Rezaei (2020) proved that NeutroAlgebra is a generalization of partial algebra and has given illustrations of NeutroAlgebras that are partial algebras. Partial algebra has some elements for which operation is undefined (not outer defined). Similarly, an AntiAlgebra is a non-empty set endowed with at least one anti operation (or anti operations) or at least one anti axiom. Examples of NeutroAlgebra and AntiAlgebra can be found in Smarandache (2020) and Smarandache and Hamidi (2020).

Examples of NeutroAlgebra and AntiAlgebra are provided in the following to make this chapter a self-contained one.

Example 1:Let 978-1-6684-3495-6.ch015.m09be the semigroup under product modulo 6. The non-trivial idempotents of 978-1-6684-3495-6.ch015.m10are 978-1-6684-3495-6.ch015.m11 (as 978-1-6684-3495-6.ch015.m12and 978-1-6684-3495-6.ch015.m13). The Cayley table for 978-1-6684-3495-6.ch015.m14under + is given in Table 1.Table 1.

Cayley table of 978-1-6684-3495-6.ch015.m15

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