M. M. Parmenter

Units in alternative loop rings

In this paper, we show that certain well known theorems concerning units in integral group rings hold more generally for integral loop rings which are alternative.

Reversible Group Rings Over Commutative Rings

Let G be a torsion group and R be a commutative ring with identity. We investigate reversible group rings RG over commutative rings, extending results of Gutan and Kisielewicz which characterize all reversible group rings over fields.

Trivial Units for Group Rings with G-adapted Coefficient Rings

For each finite group G for which the integral group ring mathbb{Z}G has only trivial units, we give ring-theoretic conditions for a commutative ring R under which the group ring RG has nontrivial units. Several examples of rings satisfying the conditions and rings not satisfying the conditions are given. In addition, we extend a well-known result for fields by showing that if R is a ring of finite characteristic and RG has only trivial units, then G has order at most 3.

Some Results on Hypercentral Units in Integral Group Rings

Abstract In this note we investigate the hypercentral units in integral group rings ℤG,where G is not necessarily torsion. One of the main results obtained is the following (Theorem 3.5): if the set of torsion elements of G is a subgroup T of G and if Z 2(𝒰) is not contained in C 𝒰(T),then T is either an Abelian group of exponent 4 or a Q* group. This extends our earlier result on torsion group rings.

Algebraic properties of necklace rings

Given a commutative ring A with 1, the necklace ring N r(A) was introduced by Metropolis and Rota [2]. In this note, we investigate the algebraic structure of necklace rings.

On *-clean group rings

A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection. Clearly a *-clean ring is clean. Vas asked whether there exists a clean ring with involution * that is not *-clean. In a recent paper, Gao, Chen and the first author investigated when a group ring RG with classical involution * is *-clean and obtained necessary and sufficient conditions for RG to be *-clean, where R is a commutative local ring and G is one of C3, C4, S3 and Q8. As a consequence, the authors provided many examples of group rings which are clean, but not *-clean. In this paper, we continue this investigation and we give a complete characterization of when the group algebra 𝔽Cp is *-clean, where 𝔽 is a field and Cp is the cyclic group of prime order p. Our main result is related closely to the irreducible factorization of a pth cyclotomic polynomial over the field 𝔽. Among other results we also obtain a complete characterization of when RCn (3 ≤ n ≤ 6) is *-clean where R is a commutative local ring.

MULTIPLICATIVE JORDAN DECOMPOSITION IN INTEGRAL GROUP RINGS OF GROUPS OF ORDER 16

If G is a finite group and K is a field of characteristic 0, then every element a in the group algebra KG has a unique additive Jordan decomposition a 1⁄4 as þ an with as; an 2 KG; as semisimple, an nilpotent and asan 1⁄4 anas. If a is a unit in KG, then as is also a unit and a has a unique multiplicative Jordan decomposition a 1⁄4 asau with au 1⁄4 1þ a 1 s an unipotent and asau 1⁄4 auas. When R is an integral domain with quotient field K and a 2 RG, it may or may not be the case that as 2 RG. If this happens to be the case for all a 2 RG (resp. for all units in RG), then we say that the additive ðresp. multiplicativeÞ Jordan decomposition holds in RG and will use the initials AJD (resp. MJD) to denote this situation. These properties have been investigated in a series of papers (1⁄21 4 ) and the previous paragraph borrows heavily from the introduction to a paper of Arora, Hales and Passi,1⁄22 where fundamental definitions and a more detailed account can be found. Of particular interest is the problem of classifying those finite groups G for which the integral group ring ZG has

Torsion Freeness of the Unit Group of 1 + Δ(G)Δ(A)*

Let ℤG be the integral group ring of a finite group G and U=U(ℤG) its group of units. For a normal subgroup N ⊲ G we denote by Δ(G, N) the kernel of the natural map ℤG → ℤ(G/N). In particular, for N = G, Δ(G, G) is the augmentation ideal Δ(G) which is the kernel of the augmentation map e. Let us denote by U 1 the units of augmentation one. It is a classical result of G. Higman that if A is an abelian group then (UℤA) is the direct product ±A × U 2 (ℤA) where U 2(ℤA) = u ∈U(ℤA) : u ≡ 1 mod (ΔA)2. Furthermore, ±A is the torsion subgroup of U(ℤA), namely, U 2(ℤA) is torsion free. An extension of this result is that if A ⊲ G and Uℤ(G/A) is trivial (equivalently G/A is abelian of exponent 2, 3, 4 or 6) then again U(ℤG) = ±G × U(1 + Δ(G)Δ(A)) with the second factor torsion free. So it was asked by Dennis [2] if the embedding ±G →U(ℤA) splits and if the normal complement is torsion free. An affirmative answer to this question also has an affirmative answer to the (ISO), the isomorphism problem. As it turns out, it was proved by Cliff-Sehgal-Weiss [1] that the answer is affirmative if G/A is abelian of odd order. The normal complement arises from ideals related to Δ(G)Δ(A). The unit group U(1 + Δ(G)Δ(A)) was proved to be torsion free for all metabelian groups G. Roggenkamp-Scott [3] answered the question of Dennis in the negative by providing counterexamples for even metabelian groups. In spite of this, the following was proposed as Problem 28 in [4].

On the regular radical of a group ring

If R is a commutative ring with identity and G a group, it is shown that the regular radical of the group ring RG is trivial if the regular radical of R is trivial. This answers in the affirmative the commutative case of a question in [4]. Further, the regular radical of RG is computed when G is an Abelian group or when G is locally finite and the order of each element of G is a unit in R.