Problems proposed by B. Gardner

The following are from the LRB. In all cases, if a universal class is not specified the question can be considered in as much generality as is feasible or desired.

  1. Are there situations in which the Kurosh lower radical construction terminates but not (necessarily) at the $\omega$ step?
  2. How many steps are required in the Kurosh lower radical construction over a class of
    (i) nil (associative) rings (Pat Stewart),
    (ii) Lie algebras,
    (iii) partially ordered rings?
  3. Find necessary conditions on a universal class for hereditary classes to define hereditary lower radical classes. Comments
  4. Let $M$ be a homomorphically closed class in some universal class such that its lower radical class $L(M)$ satisfies (ADS). If $I\lhd J\lhd A$, $I \in M$ and $I^*$ is the ideal/normal subobject of $A$ generated by $I$ , must $I^*$ be in some $M_{\alpha}$? Comments
  5. Is there a universal class in which semi-simple classes are precisely the regular classes closed under extensions and subdirect products but yet there is at least one non-hereditary semi-simple class?
  6. If in some variety as universal class a semi-simple class is hereditary, must its radical class have (ADS)? Comments
  7. If every semi-simple class is hereditary, must every radical class have (ADS)?
  8. If every class defines an upper radical class (i.e. if arbitrary intersections of semi-simple classes are semi-simple classes) must semi-simple classes be hereditary?
  9. Let $Q$ be an essentially closed regular class satisfying $(I\lhd J \lhd A \& \ J/I\in Q)\rightarrow I\lhd A$
    (i) must the upper radical class $U (Q)$ have the intersection property with respect to $Q$?
    (ii) must $U (Q)$ be hereditary? Comments
  10. (Not quite a special case of the previous problem.) Under what conditions does an essentially closed semi-simple class correspond to a hereditary radical class?
  11. When do the residually finite objects form a semi-simple class? Comments
  12. Find conditions under which strongly hereditary classes determine strict upper radical classes. Comments
  13. When are semi-simple classes closed under free products/coproducts? Comments
  14. When are radical classes closed under direct products? Direct products of $\leq \lambda$ objects for some infinite cardinal number $\lambda$? Comments

The following are not NOT from LRB.

  1. What can be said about essentially closed radical classes? Comments
  2. Radicals and Banach algebras. An old theorem of Kaplansky [5] says that all (von Neumann) regular Banach algebras are finite-dimensional. More recently, Grabiner [2] has shown that all nil Banach algebras are nilpotent. Thus it would be interesting to know which Banach algebras can be in various radical classes. Note that there are infinite-dimensional (idempotent simple and hence) hereditarily idempotent Banach algebras. The behaviour of radicals with respect to Banach algebras is interesting in other ways also. The Baer and nil radicals coincide as Dixon [1] has shown and they need not be nilpotent. The Koethe Conjecture has an affirmative answer (Grabiner [3]). Note that we are not really talking about a ``radical theory for Banach algebras'': for example the nil radical need not be a closed ideal. (This can be deduced from the results cited.) The Jacobson radical is closed and is characterized by an interesting property (topologically nil). Is the regular radical necessarily closed? Banach algebras seem to behave somewhat like a class of rings with a finiteness condition, and this may be worth pursuing: one can ask, for example, for a characterization of those radicals which coincide on Banach algebras with the nil (regular etc.) radical. Munn [5] has shown quite recently that certain Banach algebras are Brown-McCoy-semi-simple, and there are obvious generalizations of these results to consider. Workers in radicals of topological rings might find their own problems in this area.
    [1] P.G. Dixon, Semiprime Banach algebras, J. London Math. Soc. 6(1973), 676-678.
    [2] S. Grabiner, The nilpotency of Banach nil algebras, Proc. Amer. Math. Soc. 21(1969), 512.
    [3] S. Grabiner, Nilpotents in Banach algebras, J. London Math. Soc. 14(1976), 7-12.
    [4] I. Kaplansky, Regular Banach algebras, J. Indian Math. Soc. 12(1948), 57-62.
    [5] W.D. Munn, Brown-McCoy semisimplicity of certain Banach algebras, Portugaliae Math. 61(2004), 393-397.
  3. Radicals and $e$-varieties. The concept of an $e$-variety originated in semigroup theory. A regular semigroup can be given an extra unary operation by the selection of an inverse for each element. The choice is far from unique, so a given regular semigroup can have many different associated algebras of type $<2,1>$. Nevertheless there is a strong relationship between varieties of these algebras and classes of regular semigroups closed under homomorphic images, products and \emph{regular} subsemigroups. These latter classes are called $e-varieties$ or \emph{existence varieties}. For an account of this see Hall [1]. The same procedure can be followed for regular \emph{rings} and indeed $e$-varieties of regular rings with identity have been investigated [2]. In the 1970s it was shown that extension-closed varieties are radical classes (and semi-simple classes in the case of associative rings) and there are related results involving generalizations of varieties. What is the relationship between extension-closed $e$-varieties of regular rings and radicals? More specifically, can the $e$- variety concept be used to obtain useful information about radical subclasses of the regular radical class?
    [1] T.E. Hall,Identities for existence varieties of regular semigroups, Bull. Austral. Math. Soc. 40(1989), 59-77.
    [2] C. Herrmann and A. Semenova, Existence varieties of regular rings and complemented modular lattices, J. Algebra 314(2007), 235-251.
  4. Maltsev products. If $\mathcal{A}$ and $\mathcal{C}$ are classes of rings (etc.),let
    $\mathcal{A}\circ{\mathcal{C}}=\{X:(\exists{I\triangleleft{X}})(I\in{\mathcal{A}}\&X/I\in{\mathcal{C}})\}$. It is well known that if $\mathcal{A}$ and $\mathcal{C}$ are varieties, then so is $\mathcal{A}\circ{\mathcal{C}}$. Under what circumstances will $\mathcal{A}\circ{\mathcal{C}}$ be a radical class, semi-simple class, etc. when $\mathcal{A}$ and $\mathcal{C}$ are? Although as far as I'm aware these questions have not been systematically studied, there are a couple of isolated known cases in which products of radicals have been shown to be radicals: Proposition 4.6 of [1] for abelian groups (and hence for $A$-radicals); Theorem 2 (proof) in [2] for associative rings. (We have $\mathcal{J}^{k}\subseteq{\mathcal{J}\circ{\mathcal{T}_{k}}}\subseteq{\mathcal{J}\vee{\mathcal{T}_{k}}}=\mathcal{J}^{k}$.)
    [1] B.J. Gardner, Torsion classes and pure subgroups, Pacific. J. Math. 33(1970), 109-116.
    [2] N.R. McConnell and T. Stokes, Generalising quasiregularity for rings, Austral. Math. Soc. Gazette 25(1998), 250-252.