Commutative Group Rings Research Articles

Groups of Units of Finite Commutative Group Rings*

Let R be a finite commutative ring with identity and ℤ p d be the cyclic group of prime power order. Define Rℤ p d to mean the group ring of ℤ p d over R. We determine the structure of the group of units of Rℤ p d in the case when R is generated by an element whose order is not divisible by p.

Hulls of Ring Extensions

AbstractWe investigate the behavior of the quasi-Baer and the right FI-extending right ring hulls under various ring extensions including group ring extensions, full and triangular matrix ring extensions, and infinite matrix ring extensions. As a consequence, we show that for semiprime rings R and S, if R and S are Morita equivalent, then so are the quasi-Baer right ring hulls of R and S, respectively. As an application, we prove that if unital C*-algebras A and B are Morita equivalent as rings, then the bounded central closure of A and that of B are strongly Morita equivalent as C*-algebras. Our results show that the quasi-Baer property is always preserved by infinite matrix rings, unlike the Baer property. Moreover, we give an affirmative answer to an open question of Goel and Jain for the commutative group ring A[G] of a torsion-free Abelian group G over a commutative semiprime quasi-continuous ring A. Examples that illustrate and delimit the results of this paper are provided.

Maximal divisible subgroups in modular group rings of p-mixed abelian groups

The isomorphism structure of the maximal divisible subgroup of the subgroup Vp(R(G); H) Id R(G) of the normalized unit group V R(G) in a commutative group ring R(G) is completely described only in terms of R, G and H whenever R is a commutative unital ring of prime characteristic p and G is a p-mixed abelian group. In particular, the maximal divisible subgroup of V R(G) is characterized. This extends a result due to Nachev (Commun. Algebra, 1995) as well as a result due to the author (Commun. Algebra, 2010).

On the Commutative Twisted Group Algebras

Let G be an abelian group and let R be a commutative ring with identity. Denote by R t G a commutative twisted group algebra (a commutative twisted group ring) of G over R, by ℬ(R) and ℬ(R t G) the nil radicals of R and R t G, respectively, by G p the p-component of G and by G 0 the torsion subgroup of G. We prove that: i. If R is a ring of prime characteristic p, the multiplicative group R* of R is p-divisible and ℬ(R) = 0, then there exists a twisted group algebra R t 1 (G/G p ) such that R t G/ℬ(R t G) ≅ R t 1 (G/G p ) as R-algebras; ii. If R is a ring of prime characterisitic p and R* is p-divisible, then ℬ(R t G) = 0 if and only if ℬ(R) = 0 and G p = 1; and iii. If B(R) = 0, the orders of the elements of G 0 are not zero divisors in R, H is any group and the commutative twisted group algebra R t G is isomorphic as R-algebra to some twisted group algebra R t 1 H, then R t G 0 ≅ R t 1 H 0 as R-algebras.

Unit Groups of Commutative Group Rings

Let R be a commutative ring with identity. An algebraic element α over R is called an “integral algebraic element” over R if there exists a minimal polynomial of α over R which is monic. Let R be a direct product of m indecomposable rings R i , m ∈ ℕ. Denote by RG the group ring of G over R and by R* the multiplicative group of R. Let G be a finite Abelian group of exponent n and n ∈ R. In this paper we give a decomposition of RG, up to isomorphism, into a direct sum of extensions of the ring R, taking into account the number of the repetitions of these extensions. If the ring R is a field, then this result is proved in Perlis and Walker (1950). Let G be a splitting Abelian group with a torsion subgroup G 0. Denote by G p the p -component of G. We give a description of the unit group U(RG) of RG in the following cases: i. when R i is a ring of prime characteristic p i , G 0/G p i is finite and the exponent of G 0/G p i belongs to R i *; ii. when R i is of characteristic zero, R i has no nilpotents, G 0 is finite of exponent n and n ∈ R i *. For the establishment of these results we prove that if the ring R is indecomposable and n ∈ R*, then: i. the cyclotomic polynomial Φ n (x) has a unique decomposition in a product of monic irreducible factors over R; and ii. if α and β are integral algebraic elements over R which are roots of monic irreducible divisors of the cyclotomic polynomial Φ n (x), then the rings R[α] and R[β] are isomorphic.

Nilpotent Elements and Idempotents in Commutative Group Rings#

Let R be a commutative ring with identity and L an extension of R. Let α ∈ L and denote by R[α] the smallest subring of L containing R and α. In this article, we consider only extensions of kind R[α], when α is an algebraic element over R and the minimal polynomial f(x) of α is monic. In this direction, the following two problems arise in a natural way: i. If R does not contain nilpotent elements, does it follow that this property holds for R[α]? ii. If R does not have nontrivial idempotents, does it follow that it is fulfilled for R[α]. We give a positive answer to these two problems under some conditions of the polynomial f(x).

On commutative twisted group rings

Let G be an abelian group, R a commutative ring of prime characteristic p with identity and R t G a commutative twisted group ring of G over R. Suppose p is a fixed prime, G p and S(R t G) are the p-components of G and of the unit group U(R t G) of R t G, respectively. Let R* be the multiplicative group of R and let f α(S) be the α-th Ulm-Kaplansky invariant of S(R t G) where α is any ordinal. In the paper the invariants f n (S), n ∈ ℕ ∪ {0}, are calculated, provided G p = 1. Further, a commutative ring R with identity of prime characteristic p is said to be multiplicatively p-perfect if (R*)p = R*. For these rings the invariants f α(S) are calculated for any ordinal α and a description, up to an isomorphism, of the maximal divisible subgroup of S(R t G) is given.

On a decomposition formula in commutative group rings

On a decomposition formula in commutative group rings

Basic subgroups in commutative modular group rings

Let $S(RG)$ be a normed Sylow $p$-subgroup in a group ring $RG$ of an abelian group $G$ with $p$-component $G_p$ and a $p$-basic subgroup $B$ over a commutative unitary ring $R$ with prime characteristic $p$. The first central result is that $1+I(RG; B_p) + I(R(p^i)G; G)$ is basic in $S(RG)$ and $B[1+I(RG; B_p) + I(R(p^i)G; G)]$ is $p$-basic in $V(RG)$, and $[1+I(RG; B_p) + I(R(p^i)G; G)]G_p/G_p$ is basic in $S(RG)/G_p$ and $[1+I(RG; B_p) + I(R(p^i)G; G)]G/G$ is $p$-basic in $V(RG)/G$, provided in both cases $G/G_p$ is $p$-divisible and $R$ is such that its maximal perfect subring $R^{p^i}$ has no nilpotents whenever $i$ is natural. The second major result is that $B(1+I(RG; B_p))$ is $p$-basic in $V(RG)$ and $(1+I(RG; B_p))G/G$ is $p$-basic in $V(RG)/G$, provided $G/G_p$ is $p$-divisible and $R$ is perfect. In particular, under these circumstances, $S(RG)$ and $S(RG)/G_p$ are both starred or algebraically compact groups. The last results offer a new perspective on the long-standing classical conjecture which says that $S(RG)/G_p$ is totally projective. The present facts improve the results concerning this topic due to Nachev (Houston J. Math., 1996) and others obtained by us in (C. R. Acad. Bulg. Sci., 1995) and (Czechoslovak Math. J., 2002).

Constructing units in commutative group rings

Constructing units in commutative group rings

Commutative group rings of finite representation type

Let Z p denote the localization (but not the completion) of the integers at the prime p. Then, for finite groups G, the group ring Z p G has finite representation type if and only if the p-Sylow subgroups of G are cyclic of order at most p 2. In this paper, we determine the possible ranks of indecomposable Z p G-lattices for finite abelian groups G for which Z p G has finite representation type. In particular, for such groups G, we show that every indecomposable Z p G-lattice can be embedded as a sublattice of Z p G (4), but not, in general, as a sublattice of Z p G (3).

Artin Rings with Two-Generated Ideals

AbstractIt is shown that each commutative Artin local ring having each of its ideals generated by two elements is the homomorphic image of a one-dimensional local complete intersection ring which also has each of its ideals generated by two elements. It is indicated how this can be applied to show that the property that each ideal is projective over its endomorphism ring does not pass to homomorphic images, and in determining the commutative group rings with the two-generator property.

Rings with two-generated ideals

A commutative ring R is said to have the two-generator property if each ideal of R can be generated by two elements, and is said to be stable if each regular ideal of R is projective over its endomorphism ring. It is known that for a one-dimensional local Macaulay ring the two-generator property implies stability but not conversely. We extend some of the known results on rings with the two-generator property to stable rings, and determine conditions under which stable rings have the two-generator property. We also extend a structure theorem for certain finitely generated torsionfree modules over rings with the two-generator property, and some of its consequences for cancellation of direct summands, to one-dimensional rings which may not have finite integral closure, and remove the finite integral closure hypothesis from a characterization of Greither of commutative group rings with the two-generator property.

On finite generation of unit groups of commutative group rings

On finite generation of unit groups of commutative group rings

On units in commutative group rings

On units in commutative group rings

The Jacobson radical of commutative group rings

The Jacobson radical of commutative group rings

Preservation of local properties and chain conditions in commutative group rings

Preservation of local properties and chain conditions in commutative group rings

Prime ideals and localization in commutative group rings

Prime ideals and localization in commutative group rings

Quadratic forms over commutative group rings and the K-theory

Quadratic forms over commutative group rings and the K-theory