## 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

### MSC:

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