A Note on Algebras that are Sums of Two Subalgebras

摘要

We study an associative algebra A over an arbitrary field that is a sum of two subalgebras B and C (i.e., A = B + C). We show that if B is a right or left Artinian PI algebra and C is a PI algebra, then A is a PI algebra. Additionally, we generalize this result for semiprime algebras A. Consider the class of all semisimple finite dimensional algebras A = B + C for some subalgebras B and C that satisfy given polynomial identities f = 0 and g = 0, respectively. We prove that all algebras in this class satisfy a common polynomial identity.