Group rings

Group rings and group algebras: unit and zero-divisor questions (Kaplansky unit conjecture), Zassenhaus conjectures, augmentation and dimension subgroups, Ore conditions, Jordan decomposition, projective modules over group rings.

• Dimension subgroups
• Group-ring powers in fields
• Inversion on commutative group rings
• Jordan decomposition in integral group rings
• Kaplansky's unit conjecture
• Ore condition for group algebras
• Projective modules over group rings
• Right ideals of free group algebras
• Units of integral group rings
• Zassenhaus conjectures

DefinitionAugmentation right ideal of a subset#✓fully proved

Let \(G\) be a group, \(S \subseteq G\) a subset and \(\mathbb{Z}G\) the integral group ring of \(G\). This is the right ideal of \(\mathbb{Z}G\) generated by the elements \(g - 1\) with \(g \in S\), regarded as a right \(\mathbb{Z}G\)-submodule of \(\mathbb{Z}G\).

noncomputable def augmentationRightIdeal {G : Type*} [Group G] (S : Set G) :
    Submodule (MonoidAlgebra ℤ G)ᵐᵒᵖ (MonoidAlgebra ℤ G) :=
  Submodule.span _ ((fun g => MonoidAlgebra.of ℤ G g - 1) '' S)

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