On the radical theory of involution algebras. (English) Zbl 0615.16008

In the variety of involution K-algebras over a commutative ring K with 1, a subclass \({\mathfrak S}\) is the semisimple class of a Kurosh-Amitsur radical, if \({\mathfrak S}\) is regular, coinductive, closed under extensions and the class \({\mathcal S}{\mathcal U}{\mathfrak S}\) is hereditary where \({\mathcal S}\) and \({\mathcal U}\) denote the semisimple and upper radical operator, respectively. Let \({\mathfrak C}\) be a subclass of involution algebras satisfying condition (ID): \(A^{id}\in {\mathfrak C}\) if and only if \(A^{- id}\in {\mathfrak C}\) whenever \(A^ 2=0\); if \({\mathfrak C}\) is regular, then \({\mathcal U}{\mathfrak C}\) satisfies A-D-S, and if \({\mathfrak C}\) is homomorphically closed, then its lower radical \({\mathcal L}{\mathfrak C}\) satisfies A-D-S (which means that the radical of any involution ideal is an involution ideal in the involution algebra). Let \({\mathfrak F}\) be a regular, coinductive subclass which is closed under extensions. Then the following are equivalent: i) \({\mathcal U}{\mathfrak F}\) satisfies A-D-S, ii) \({\mathfrak F}\) satisfies (ID), iii) if \(A^*\in {\mathfrak F}\) and \(A^ 2=0\), then \(A^{\circ}\in {\mathfrak F}\) for every involution \(\circ\), iv) if \(A^*\in {\mathfrak F}\) and \(A^ 2=0\), then \(A^{-*}\in {\mathfrak F}\), v) if \(A^*\in {\mathfrak F}\) and A is nilpotent, then every nilpotent involution algebra built on A is in \({\mathfrak F}\). An example is given showing that a coradical class need not be a semisimple class.
Reviewer: R.Wiegandt


16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16Nxx Radicals and radical properties of associative rings
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