From the text: “We obtain nonassociative extensions of some basic results in the associative PI and GPI theories, such as Regev’s tensor product PI-theorem, Kaplansky’s primitive PI-theorem, Posner’s prime PI-theorem, Amitsur’s primitive GPI-theorem, and Martindale’s prime GPI-theorem.”
“Section 1 is devoted to the study of algebras with PI multiplication algebra, providing nonassociative extensions of the classical theorems by Regev on tensor products of associative PI-algebras, by Kaplansky on primitive associative PI-algebras, and by Posner on prime associative PI-algebras.
Section 2 discusses multiplicatively prime (m.p.) algebras with GPI multiplication algebra, and contains associative and nonassociative GPI-versions of the Regev theorem. Moreover, we provide nonassociative extensions of the Amitsur theorem on primitive associative GPI-algebras and of the Martindale prime GPI-theorem.
Section 3 is devoted to complete the nonassociative Martindale and Posner theorems obtained in sections 2 and 1, respectively. Our development relies on the introduction of a reasonable notion of multiplicative generalized polynomial (in short MGP) for m.p. algebras. Roughly speaking, for a given m.p. algebra \(A\), a MGP is a finite sum of monomials of the form \(F_1X_{i_1}F_2X_{i_2}\cdots F_nX_{i_n}(q)\) for \(n\in\mathbb N\), \(q\in Q_A\), \(F_1,\ldots,F_n\in M_{C_A}(Q_A)\), and \(X_{i_1}, X_{i_2},\ldots,X_{i_n}\) noncommutative formal variables. We say that \(M(A)\) satisfy a MGP \(\Phi = \Phi(X_1,\ldots,X_n)\), or that \(\Phi\) is a MGPI on \(M(A)\), if \(\Phi(T_1,\ldots,T_n)\) for all \(T_1,\ldots,T_n\in M(A)\). As a matter of fact, the existence of a nonzero MGPI on \(M(A)\) is equivalent to the fact that \(M(A)\) is GPI.”