Generalized polynomial identities

Louis Halle Rowen
1975 Journal of Algebra  
A ring R is said to satisfy a generalized polynomial identity if some polynomial C rilXilris ... ritXiTi,t+l vanishes for all evaluations in R obtained by specializing the indeterminates to elements of R. Interesting results have been obtained by Amitsur [l], Martindale [IO] and Jain [8], based on this seemingly mild condition, but this theory has been hindered by the dependence of definitions of admissible generalized polynomial identities on the extended centroid; this incumberance has
more » ... d results, complicated proofs, and rendered completely impossible the study of generalized polynomial identities of non-prime rings. In Section 1, we give an intrinsic definition of a proper generalized polynomial identity which leads to the establishment of a natural varietal framework. In Section 2, new, reasonably easy proofs are given of major results of Amitsur, Martindale and Jain characterizing primitive and prime rings with generalized identities; the assumptions made in the hypotheses of these theorems boils down to assuming the identities are proper. A crude extension to arbitrary rings is given in Section 3, with the conclusion that the upper and lower nilradicals coincide in suitable situations, and the results of Sections l-3 are put in the context of rings with involution to yield a straightforward proof of a major theorem of Herstein-Martindale-Amitsur; a generalization of this result is then given in the GI theory. This exposition is essentially independent of previous works on generalized identities, utilizing only the more straightforward results of [l] and [lo].
doi:10.1016/0021-8693(75)90170-2 fatcat:r3rclw67kvc7hda7zflzc6odyi