Structure theorems are obtained for some classes of modules over the ring of pseudorational numbers and some classes of quotient divisible mixed groups.

background

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Infinite Abelian groups

Modules over the ring of pseudorational numbers and quotient divisible groups

Abstract. Recall the well-known notion of a cellular cover e:HG$e: H \\longrightarrow G$ from algebraic topology (here for groups and R$R$-modules). The map e$e$ is a homomorphism such that any homomorphism :HG$\\phi : H \\longrightarrow G$ factors uniquely through e$e$. The existence of cellular covers of groups and R$R$-modules was shown in [14, 31] and a more explicit construction was given in [8, Section 6.3]. We want to base the existence and uniqueness of cellular covers on classical results in group and module theory; see Baer's [2, 3, 4] and Dickson's [13]. With the help of an argument from [8] (partly from [14]) we shift the result from R$R$-modules to the existence of cellular covers over any variety ${\\mathcal {V}}$ of groups and get the following result. If G$G$ belongs to some variety ${\\mathcal {V}}$, then any cellular cover 1KCG$1\\longrightarrow K\\longrightarrow C\\longrightarrow G$ is a short exact sequence with C c $C\\in {\\mathcal {V}}^c$ (and hence K,C,G c $K,C,G\\in {\\mathcal {V}}^c$). Here c ${\\mathcal {V}}^c$ is the variety generated by all central extensions of groups V/V$V/{\\mathfrak {z}}V$ with V$V\\in {\\mathcal {V}}$. This unifies older arguments (from [14, 8]) and adds 2 0 $2^{{\\aleph _0}}$ new group varieties. Moreover, the following interesting problem arrives naturally. Can we find group varieties which are not closed under taking cellular covers? The positive answer to this question needs work and application of deep results from combinatorial group theory and on the Schur multiplier; see [29, 34]. We will show that for each prime p>10 75 $p > 10^{75}$ the Burnside-variety p ${\\mathcal {B}}_p$ satisfying x p =1$x^p =1$ has cellular covers with free abelian subgroups of infinite rank, thus surely these cellular covers of groups from p ${\\mathcal {B}}_p$ are outside p ${\\mathcal {B}}_p$.

Cellular covers for R$R$-modules and varieties of groups

Computability of certain classes of constructive algebras

We prove for a large family of rings R that their lambda-pure global dimension is greater than one for each infinite regular cardinal lambda. This answers in negative a problem posed by Rosicky. The derived categories of such rings then do not satisfy the Adams lambda-representability for morphisms for any lambda. Equivalently, they are examples of well generated triangulated categories whose lambda-abelianization in the sense of Neeman is not a full functor for any lambda. In particular we show that given a compactly generated triangulated category, one may not be able to find a Rosicky functor among the lambda-abelianization functors.

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On the abelianization of derived categories and a negative solution to Rosicky's problem

In a recent paper [4] S. Feigelstock considers (not necessarily associative) rings the ideal lattices of which are totally ordered. He calls such rings TOLI rings. Clearly, if Ris a TOLI ring, the lattice of fully invariant subgroups of its additive group R+ must be totally ordered, too. In this note we consider abelian groups A which possess this latter property.

Infinite Abelian groups

Citations

Modules over the ring of pseudorational numbers and quotient divisible groups

Cellular covers for R$R$-modules and varieties of groups

Computability of certain classes of constructive algebras

On the abelianization of derived categories and a negative solution to Rosicky's problem

Abelian groups which are uniserial as modules over their endomorphism rings

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