On Non-Associative Rings

Ida Kurnia Waliyanti,Indah Emilia Wijayanti,M Farchani Rosyid

doi:10.13189/ms.2021.090212

Abstract

Jordan ring is one example of the non-associative rings. We can construct a Jordan ring from an associative ring by defining the Jordan product. In this paper, we discuss the properties of non-associative rings by studying the properties of the Jordan rings. All of the ideals of a non-associative ring R are non-associative, except the ideal generated by the associator in R. Hence, a quotient ring <img src=image/13422591_01.gif> can be constructed, where <img src=image/13422591_02.gif> is the ideal generated by associators in R. The fundamental theorem of the homomorphism ring can be applied to the non-associative rings. By a little modification, we can find that <img src=image/13422591_01.gif> is isomorphic to <img src=image/13422591_03.gif>. Furthermore, we define a module over a non-associative ring and investigate its properties. We also give some examples of such modules. We show if M is a module over a non-associative ring R, then M is also a module over <img src=image/13422591_01.gif> if <img src=image/13422591_02.gif> is contained in the annihilator of R. Moreover, we define the tensor product of modules over a non-associative ring. The tensor product of the modules over a non-associative ring is commutative and associative up to isomorphism but not element by element.

Highlights

In the development of the ring theory, there has been much discussion about the associative ring
We discuss the properties of non-associative rings by studying the properties of the Jordan rings
The fundamental theorem of the homomorphism ring can be applied to the non-associative rings

Summary

Introduction

In the development of the ring theory, there has been much discussion about the associative ring. If A is commutative, a Jordan ring formed is associative, but the converse is not always true. The explanation of such property and examples will be given in the second section. After explaining the Jordan ring, we define the ideals of the non-associative ring. Carotenuto [3, 4] has given a different definition and called it the Jordan module. This definition began from the understanding of Jordan algebra. A Jordan module is one example of the modules over a nonassociative ring, as defined in 2.10. We conduct the tensor product of modules over the non-associative ring RJ

Non-Associative Rings

Module over a Non Associative Ring

Tensor Module Over Jordan Ring RJ

Conclusions