Chapter 1 is a short review of the main results in some areas of the theory of group rings. In the first half of Chapter 2 we determine the ideal theoretic structure of the group ring RG where G is the direct product of a finite Abelian group and an ordered group with R a completely primary ring. Our choice of rings and groups entails that the study centres mainly on zero divisor ideals of group rings and hence it contributes in a small way to the zero divisor problem. We show that if R is a completely primary ring, then there exists a one-one correspondence of the prime zero divisor ideals in RG and RG¯, G finite cyclic of order n. If R is a ring with the property α, β € R, then αβ = 0 implies βα = 0, and S is an ordered semigroup, we show that if ∑α¡s¡ ∈ RS is a divisor of zero, then the coefficients α¡ belong to a zero divisor ideal in R. The converse is proved in the case where R is a commutative Noetherian ring. These results are applied to give an account of the zero divisors in the group ring over the direct product of a finite Abelian group and an ordered group with coefficients in a completely primary ring. In the second half of Chapter 2 we determine the units of the group ring RG where R is not necessarily commutative and G is an ordered group. If R is a ring such that if α, β € R and αβ = 0, then βα = 0, and if G is an ordered group, then we show that ∑αg(subscript)g is a unit in RG if and only if there exists ∑βh(subscript)h in RG such that∑αg(subscript)βg(subscript)-1 = 1 and αg(subscriptβh is nilpotent whenever GH≠1. We also show that if R is a ring with no nilpotent elements ≠0 and no idempotents ≠0,1, then RG has only trivial units. In this chapter we also consider strongly prime rings. We prove that RG is strongly prime if R is strongly prime and G is an unique product (u.p.) group. If H ⊲ G such that G/H is right ordered, then it is shown that RG is strongly prime if RH is strongly prime. In Chapter 3 results are derived to indicate the relations between certain classes of ideals in R and RG. If δ is a property of ideals defined for ideals in R and RG, then the "going up" condition holds for δ-ideals if Q being a δ-ideal in R implies that QG is a δ-ideal in RG. The "going down" condition is satisfied if P being a δ-ideal in RG implies that P∩ R is a δ-ideal in R. We proved that the "going up" and "going down" conditions are satisfied for prime ideals, ℓ-prime ideals, q-semiprime ideals and strongly prime ideals. These results are then applied to obtain certain relations between different radicals of the ring R and the group ring (semigroup ring) RG (RS). Similarly, results about the relation between the ideals and the radicals of the group rings RH and RG, where H is a central subgroup of G, are obtained. For the upper nil radical we prove that ⋃(RG) (RH) ⊆ RG, H a central subgroup of G, if G/H is an ordered group . If S is an ordered semigroup, however, then ⋃(RS) ⊆ ⋃(R)S for any ring R. In Chapter 4 we determine relations between various radicals in certain classes of group rings. In Section 4.3, as an extension of a result of Tan, we prove that P(R)G = P(RG) , R a ring with identity , if and only if the order of no finite normal subgroup of G is a zero divisor in R/P(R). If R is any ring with identity and H a normal subgroup of G such that G/H is an ordered group, we show that ⊓(RH)·RG = ⋃(RG) = ⊓(RG) , if ⋃(RH) is nilpotent. Similar results are obtained for the semigroup ring RS, S ordered. It is also shown if R is commutative and G finite of order n, then J(R)G = J(RG) if and only if n is not a zero divisor in R/J(R), J(R) being the Jacobson radical of R. For the Brown HcCoy radical we determine the following: If R is Brown McCoy semisimple or if R is a simple ring with identity, then B(RG) = (0), where G is a finitely generated torsion free Abelian group. In the last section we determine further relations between some of the previously defined radicals, in particular between P(R), U(R) and J(R). Among other results, the following relations between the abovementioned radicals are obtained: U(RS) = U(R)S = P(RS) = J(RS) where R is a left Goldie ring and S an ordered semigroup with unity

In this thesis we will be exclusively considering uncountable groups and semigroups. Roughly speaking the underlying problem is to find “large” subgroups (or subsemigroups) of the object in question, where we consider three different notions of “largeness”: we classify all the subsemigroups of the set of all mapping from a countable set back to itself which contains a specific uncountable subsemigroup; we investigate topological “largeness”, in particular subgroups which are finitely generated and dense; we investigate if it is possible to find an integer r such that any countable collection of elements belongs to some r-generated subsemigroup, and more precisely can these elements be obtained by multiplying the generators in a prescribed fashion.

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This thesis is broadly concerned with two problems: investigating the ergodic properties of boundary actions, and investigating various properties of Schreier graphs. Our main result concerning the former problem is that, in a variety of situations, the action of an invariant random subgroup of a group G on a boundary of G (e.g. the hyperbolic boundary, or the Poisson boundary) is conservative (there are no wandering sets). This addresses a question asked by Grigorchuk, Kaimanovich, and Nagnibeda and establishes a connection between invariant random subgroups and normal subgroups. We approach the latter problem from a number of directions (in particular, both in the presence and the absence of a probability measure), with an emphasis on what we term Schreier structures (edge-labelings of a given graph which turn it into a Schreier coset graph). One of our main results is that, under mild assumptions, there exists a rich space of invariant Schreier structures over a given unimodular graph structure, in that this space contains uncountably many ergodic measures, many of which we are able to describe explicitly.

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A group Γ is of type F_n for some n ≥ 1 if it has a classifying complex with finite n-skeleton. These properties generalise the classical notions of finite generation and finite presentability. We investigate the higher finiteness properties for fibre products of groups.

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