What are the differences between rings, groups, and fields?
A ring is an abelian group with an additional operation, where the second operation is associative and the distributive property make the two operations "compatible". A field is a ring such that the second operation also satisfies all the properties of an abelian group (after throwing out the additive identity), i.e. it has multiplicative ...
What does the group ring - Mathematics Stack Exchange
Apr 29, 2017 · You might consult Chapter 9 in Milies, Sehgal, "An Introduction to Group Rings". It discusses how several invariants of groups are encoded in their integral group rings, and how this leads to an affirmative answer of the integral group ring isomorphism problem for …
Group Acting on a Ring - Mathematics Stack Exchange
Apr 21, 2015 · What would be the definition for a group action on a ring? I could not find one online. Would this be acceptable? A group action of a group G on a ring R is a map from G x R to R defined by g(r)=g.r satisfying the following properties: (1) g.(g'.r)=(g.g').r for all g,g'ϵ G, r ϵ R (2) 1.r=r for all r ϵ R
Center of a group ring? - Mathematics Stack Exchange
Oct 3, 2020 · The centre of a group ring is spanned by sums of elements of conjugacy classes. Suppose that C ⊆ G C ⊆ G is a conjugacy class, and set sC =∑x∈C x s C = ∑ x ∈ C x, then we have gsCg−1 = sC g s C g − 1 = s C, so certainly sC s C is central in the group algebra. Conversely, suppose that ∑x∈Grxx ∑ x ∈ G r x x is central in ...
abstract algebra - Module and group ring: definitions and …
The group ring R[G] is the ring of polynomials with coefficients in R and variables coming from G (which explains the notation). However, we define the multiplication of two polynomials using the multiplication of our group. As an example, consider Z[C3] where C3 = g | g3 = 1 is the cyclic group with 3 elements.
linear algebra - Is there a relationship between vector spaces and ...
Sep 8, 2015 · The ring of matrices has a multiplicative identity element, but this is not required by definition. There are rings without a multiplicative identity. So a ring is a group with an additional structure - that obtained by another operation. A field is a very special kind of ring.
Units of a group ring. - Mathematics Stack Exchange
5. For a general group G G this is a hard problem. Let me mention the partial progress that I remember. For one thing, if G G isn't torsion-free, there is a theorem in Passman's Algebraic structure of group rings that says that the units of F[G] F [G] for a field F F are trivial iff G G is F2[C2] F 2 [C 2], F2[C3] F 2 [C 3] or F3[C2] F 3 [C 2].
Automorphisms of group ring - Mathematics Stack Exchange
Mar 21, 2024 · The augmentation is the sum of the coefficients and an automorphism is augmented, if it respects the augmentation. By a result of Higman, one of the first theorems on units in group rings, if A A is a finite abelian group, then the only units of finite order and augmentation 1 are the elements of A A. Hence, if you assume your ϕ ϕ to be ...
abstract algebra - When is a group ring an integral domain ...
When is a group ring an integral domain. If R R is an integral domain (I am having Z Z or a field in mind) and G G a (not necessarily finite) group then we can form the group ring R(G) R (G). Note that if gn+1 = e g n + 1 = e then (e − g)(e + g … +gn) = e −gn+1 = 0 (e − g) (e + g … + g n) = e − g n + 1 = 0. This means if G G has ...
In $\mathbb{Z}[S_3]$, the group ring of the symmetric group on 3 ...
Oct 12, 2020 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.