Study of Some Degree Five Identities of Type (4,1)

In non-associative algebra, irreducible identities of degree five are the least studied. Following Osborn s studies, the only identity of type (5) has generated very little literature, as seen in ([1]) and ([2]). Hence, our interest in identities of the following type. The purpose of this study is to enable us to consider a baric case study at a later stage, such as in ([3]) and ([4]). This paper is devoted to the study of three of type (4,1), taken from the families of irreducible degree five identities of Osborn. We conduct this study in the presence of an idempotent, through a Peirce decomposition, depending on whether the Peirce polynomial is reducible or not over the base field of the algebra. In each studied case, we find two orthogonal subalgebras. Therefore, in the first two studied identities, we manage to show that there is a homomorphism over one of the subspaces of A, whose kernel is an ideal.