The central polynomials of the infinite-dimensional unitary and nonunitary Grassmann algebras

Abstract

For a fixed field $k$, let $k_0\langle X\rangle$ and $k_1\langle X \rangle$ denote respectively the free nonunitary associative $k$-algebra and the free unitary associative $k$-algebra on the countable set $X=\{x_1, x_2, \ldots\}.$ A polynomial $f\in k_i\langle X\rangle$ is called a central polynomial for an associative algebra $A$ if every evaluation of $f$ on $A$ lies in the center of $A.$ The set of all central polynomials of $A$ is a $T$-space of $k_i\langle X\rangle,$ i.e, a subspace closed under all endomorphisms of $k_i\langle X\rangle.$ In this paper the authors describe the T-space of central polynomials for both the unitary and the nonunitary infinite-dimensional Grassmann algebra over a field of characteristic $p\neq 2.$ Similar results were obtained in [A. P. Brandão Jr., P. Koshlukov, A. Krasilnikov and É. Alves da Silva, Israel J. Math. 179 (2010), 127–144; MR2735036].