Free commutative quasi-regular algebras and algebras without quasi-regular subalgebras

Prime rings for which the set of nonzero ideals is a special class

“…for all i £ I. This latter condition requires that / n Ker(/z) = 0, whence, B being a prime ring, Ker(h) = 0 and B is isomorphic to an accessible subring of A and hence, by (2), an ideal of A.…”

“…Since a class J[ of prime rings is special if and only if it is hereditary with respect to nonzero ideals and closed under essential extensions (see [3] for this characterization, which differs slightly from the original definition given by Andrunakievich [1]) it is clear that every nonempty intersection of special classes is special, and in particular every prime ring A generates a special class (which, following [6] and [2], we shall call n A ). This n A is the intersection of all special classes containing A .…”

On Bad Supernilpotent Radicals

“…Examples of nonspecial supernilpotent radicals were given in [3,4,6,7,12,14,15]. Since a supernilpotent radical α is special if and only if α = U(π(α)) [2,8], nontrivial bad supernilpotent radicals provide the most natural counterexamples to Andrunakievich's question. The first such example was constructed by Ryabukhin [11] who showed that the upper radical generated by the class of all Boolean rings which do not contain an ideal which is a prime field with two elements is a supernilpotent but nonspecial radical.…”

“…A ring A is called Boolean if a 2 = a for every a ∈ A. The fundamental definitions and properties of radicals can be found in [2] and [8]. A class µ of rings is called hereditary if µ is closed under ideals.…”

On questions of B. J. Gardner and A. D. Sands

“…For the basic notions and results we refer to [2,4,10,11]. Let &> be any class of rings and =2 any class of prime rings.…”