### Journals Information

**
Mathematics and Statistics Vol. 9(2), pp. 172 - 178 DOI: 10.13189/ms.2021.090212 Reprint (PDF) (369Kb) **

## 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.