Mathematics and Statistics Vol. 9(2), pp. 172 - 178
DOI: 10.13189/ms.2021.090212
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On Non-Associative Rings

Ida Kurnia Waliyanti ^1,2,*, Indah Emilia Wijayanti ¹, M. Farchani Rosyid ^3
1 Department of Mathematics, Universitas Gadjah Mada, Yogyakarta, 55281, Indonesia
² Department of Mathematics Education, Universitas Khairun, Ternate, 97719, Indonesia
³ Department of Physics, Universitas Gadjah Mada, Yogyakarta, 55281, Indonesia

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 can be constructed, where 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 is isomorphic to . 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 if 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.

KEYWORDS
Non-Associative Ring, Jordan Ring, Ideal, Module, Tensor Product

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Ida Kurnia Waliyanti , Indah Emilia Wijayanti , M. Farchani Rosyid , "On Non-Associative Rings," Mathematics and Statistics, Vol. 9, No. 2, pp. 172 - 178, 2021. DOI: 10.13189/ms.2021.090212.

(b). APA Format:
Ida Kurnia Waliyanti , Indah Emilia Wijayanti , M. Farchani Rosyid (2021). On Non-Associative Rings. Mathematics and Statistics, 9(2), 172 - 178. DOI: 10.13189/ms.2021.090212.