Ideals in the center of symmetric algebras

Abstract

We study symmetric algebras A over an algebraically closed field F in which the Jacobson radical of the center of A, the socle of the center of A or the Reynolds ideal of A are ideals.Let F be an algebraically closed field and let A be a finite-dimensional F -algebra. As customary, we denote its center by Z(A), its Jacobson radical by J(A) and its (left) socle, the sum of all simple left ideals of A, by soc(A). In this paper, we are interested in the Jacobson radical J(Z(A)) and the socle soc(Z(A)) of Z(A) as well as the Reynolds ideal R(A) = soc(A) ∩ Z(A) of A. All three subspaces are ideals in Z(A). We study the following properties:In this paper, an ideal I of A is always meant to be a two-sided ideal of A and we denote it by I A. Note that the property (P1) is equivalent to A • J(Z(A)) ⊆ J(Z(A)), and the corresponding statement holds for (P2) and (P3). The properties (P1) -(P3) are trivially satisfied whenever A is commutative. Thus we can view these conditions as weak commutativity properties.The question (P1) has already been answered for group algebras and their p-blocks by Clarke [5], Koshitani [8] and Külshammer [11]. The latter paper additionally contains some results on arbitrary symmetric algebras. Moreover, Landrock [12] has proven that J(Z(A)) is an ideal of A if A is a symmetric local algebra of dimension at most ten.