e-Symmetric rings

Let [Formula: see text] be a ring and [Formula: see text] an idempotent of [Formula: see text], [Formula: see text] is called an [Formula: see text]-symmetric ring if [Formula: see text] implies [Formula: see text] for all [Formula: see text]. Obviously, [Formula: see text] is a symmetric ring if and only if [Formula: see text] is a [Formula: see text]-symmetric ring. In this paper, we show that a ring [Formula: see text] is [Formula: see text]-symmetric if and only if [Formula: see text] is left semicentral and [Formula: see text] is symmetric. As an application, we show that a ring [Formula: see text] is left min-abel if and only if [Formula: see text] is [Formula: see text]-symmetric for each left minimal idempotent [Formula: see text] of [Formula: see text]. We also introduce the definition of strongly [Formula: see text]-symmetric ring and prove that [Formula: see text] is a strongly [Formula: see text]-symmetric ring if and only if [Formula: see text] and [Formula: see text] is a symmetric ring. Finally, we introduce [Formula: see text]-reduced ring and study some properties of it.

The Galois extensions induced by idempotents in a Galois algebra

LetBbe a Galois algebra with Galois groupG,Jg={b∈B|bx=g(x)b for all x∈B}for eachg∈G,egthe central idempotent such thatBJg=Beg, andeK=∑g∈K,eg≠1egfor a subgroupKofG. ThenBeKis a Galois extension with the Galois groupG(eK)(={g∈G|g(eK)=eK})containingKand the normalizerN(K)ofKinG. An equivalence condition is also given forG(eK)=N(K), andBeGis shown to be a direct sum of allBeigenerated by a minimal idempotentei. Moreover, a characterization for a Galois extensionBis shown in terms of the Galois extensionBeGandB(1−eG).

Spectral characterization of the socle in Jordan–Banach algebras

If A is a complex Banach algebra the socle, denoted by Soc A, is by definition the sum of all minimal left (resp. right) ideals of A. Equivalently the socle is the sum of all left ideals (resp. right ideals) of the form Ap (resp. pA) where p is a minimal idempotent, that is p2 = p and pAp = ℂp. If A is finite-dimensional then A coincides with its socle. If A = B(X), the algebra of bounded operators on a Banach space X, the socle of A consists of finite-rank operators. For more details about the socle see [1], pp. 78–87 and [3], pp. 110–113.

Idempotent Ideals in Perfect Rings

All rings considered in this note have an identity element, and all R-modules are unitary.Bass (2) defined a left perfect ring as a ring R satisfying the minimum condition on principal right ideals. A commutative ring R is perfect if and only if R is a direct sum of finitely many local rings whose radicals are T-nilpotent. Therefore, the commutative perfect rings with finite global projective dimension are just the direct sums of finitely many commutative fields, and hence they trivially satisfy the minimum condition for all ideals. However, in the non-commutative case, even hereditary perfect rings are not necessarily right or left artinian (cf. Example 3.4).Each left perfect ring R has only finitely many idempotent (two-sided) ideals (Corollary 2.3), where the ideal X of R is called idempotent, if X = X2. Hence, it makes sense to consider minimal idempotent idealsof the left perfect ring R, i.e., ideals of R which are minimal in the set of all idemponent ideals of R.