This book has been written about associative algebras. However, there is an analogous “PI-theory” that can be carried out for nonassociative rings, which is outlined here.
Source: wiktionary
Nonassociative
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1996, E. G. Goodaire, E. Jespers, C. Polcino Milies, Alternative Loop Rings, Elsevier, page 5, Two important functions in nonassociative ring theory are the commutator and associator which, for elements a, b, c in a ring are defined respectively by [a, b] = ab − ba and [a, b, c] = a(bc) − (ab)c.
Source: wiktionary
2012, W. B. Vasantha Kandasamy, Florentin Smarandache, Non Associative Algebraic Structures Using Finite Complex Numbers, Zip Publishing, page 5, Authors in this book for the first time have constructed nonassociative structures like groupoids, quasi loops, non associative semirings and rings using finite complex modulo integers.
Source: wiktionary
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