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A ring is nil clean if each of its elements is the sum of an idempotent and a nilpotent. In Sahinkaya et al. [13], it was shown that, for a ring R and a symmetric group S 3, the group ring RS 3 is ...

A group G is said to be a B(n, k) group if for any n-element subset A of G, |A2| ≤k. In this paper, a characterization of B(5, 18) groups is given. It is shown that G is a B(5, 18) group if and only if one of the following statements holds: (1) G is abelian; (2) |G| ≤18; (3) G ≅ ⟨ a, b | a5 = b4 = 1, ab = a−1 ⟩.

A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection (*-invariant idempotent). Recently, Gao, Chen, and Li 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. Most recently, Li, Yuan, and Parmenter gave a complete characterization of when the group algebra FCp is *-clean, where F is a field and Cp is a cyclic group of prime order p. In this article, we extend the above mentioned result from FCp to FqCpk, where Fq is a finite field and Cpk is a cyclic group of an odd prime power order pk. For the general case when G = Cn is cyclic group of order n, we also provide a necessary condition and a few sufficient conditions for FqCn to be *-clean.

Let R be a commutative ring with 1 ≠ 0, G be a nontrivial finite group, and let Z(R) be the set of zero divisors of R. The zero-divisor graph of R is defined as the graph Γ(R) whose vertex set is Z(R)* = Z(R)∖{0} and two distinct vertices a and b are adjacent if and only if ab = 0. In this paper, we investigate the interplay between the ring-theoretic properties of group rings RG and the graph-theoretic properties of Γ(RG). We characterize finite commutative group rings RG for which either diam(Γ(RG)) ≤2 or gr(Γ(RG)) ≥4. Also, we investigate the isomorphism problem for zero-divisor graphs of group rings. First, we show that the rank and the cardinality of a finite abelian p-group are determined by the zero-divisor graph of its modular group ring. With the notion of zero-divisor graphs extended to noncommutative rings, it is also shown that two finite semisimple group rings are isomorphic if and only if their zero-divisor graphs are isomorphic. Finally, we show that finite noncommutative reversible group r...

A group G is said to be a B(n, k) group if for any n-element subset {a 1,…, a n } of G, |{a i a j | 1 ≤ i, j ≤ n}| ≤k. In this article, we give a complete characterization of B(4, 13) 2-groups, and then obtain a complete characterization of B(4, 13) groups.

Let k ≥ 2. A ring R is called a B k -ring if for all k-subsets K. All commutative rings are B k -rings, hence it is natural to seek conditions which imply that a B k -ring is commutative. We present some commutativity results for arbitrary k, and then focus on B 4-rings and B 5-rings with 1. One of our results asserts that, if R is a finite 2-torsion-free B 4-ring or B 5-ring with 1, then R is commutative.

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.

Let Z n (𝒰) denote the n'th term of the upper central series of the unit group 𝒰 of ℤG and [Ztilde](𝒰) = Z n (𝒰). It is shown that if the set of torsion elements of G forms a subgroup T and [Ztilde](𝒰)⊈C 𝒰(T), then T is either an Abelian 2-group or a Q-group. Moreover, [Ztilde](𝒰) ⊆ G ⋅ C 𝒰(T) whenever G is an FC group. #Communicated by M. Ferrero.

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.

Let G be an arbitrary group. For any nontrivial bicyclic unit u ∈ G, a necessary and sufficient condition for which the pair {u, uf } generates a nonabelian free group is given. Moreover, it is proved that the above pair always generates a torsionfree subgroup of the unit group U( G) and the structure of this subgroup ⟨u, uf ⟩ is characterized.

Let R be a ring. For two fixed positive integers m and n, a right R-module M is called (m, n)-injective if every right R-homomorphism from an n-generated submodule of Rm to M extends to one from Rm to M. This definition unifies several definitions on generalizations of injectivity of modules. The aim of this paper is to investigate properties of the (m, n)-injective modules. Various results are developed, many extending known results.

In this paper, we are mainly interested in describing constructively u(Z(G × C 2)), where u(ZG) has been described in some way. We first consider unitary units and prove that if u(ZG) is generated by unitary units, then U(Z(G × C2)) is also generated by unitary units. Then we consider bicyclic units and ask the following question: If G has a normal complement generated by bicyclic units, does G × C 2 also have a normal complement generated by bicyclic units? We give a negative answer to the above in general, showing that none of the normal complements of D 8 × C 2 × C 2 is generated by bicyclic units, although a normal complement of D 8 × C 2 is indeed generated by bicyclic units.

In this paper, we investigate the properties of the normalizers of the unitary subgroup uf(ZG) in an integral group rings. One of our main results is Theorem 2.6 which character¬izes the second normalizer of the unitary subgroup. As a conse¬quence of this theorem, we prove that the second normalizer of uf(ZG) coincides with the first normalizer when G is a periodic group. Among other results, we give necessary and sufficient conditions for which the unitary subgroup is normal in the unit group when G is periodic and also characterize when all bicyclic units are nontrivial and elements of the normalizer of the unitary subgroup.