WARFIELD INVARIANTS IN COMMUTATIVE GROUP RINGS

In this study, we construct a new public key cryptosystem both like El-Gamal and RSA cryptosystem based on Bass cyclic units in integral group rings. Naturally, the underlying hard problems for this system are both decisional Diffie-Hellman assumption and factorization of integers. Since the system is related to RSA scheme indirectly, key generation process is very efficient. This cryptosystem actually supports and strengthens the current public key cryptosystems in [6], [8] and [20]. Encryption algorithm in the system is constructed via non-commutative operations in integral group rings of dihedral groups. Finally, we briefly discuss the security of our cryptosystem against some known attacks.

Let R be a ring with identity of characteristic two and G a nontrivial torsion group. We show that if the units in the group ring $RG$ are all trivial, then G must be cyclic of order two or three. We also consider the case where R is a commutative ring with identity of odd prime characteristic and G is a nontrivial locally finite group. We show that in this case, if the units in $RG$ are all trivial, then G must be cyclic of order two. These results improve on a result of Herman et al. [‘Trivial units for group rings with G-adapted coefficient rings’, Canad. Math. Bull.48(1) (2005), 80–89].

In 1963, Gilmer characterized all finite commutative rings with a cyclic group of units and, in 1967, Eldridge and Fischer generalized these results to rings with minimum condition. In the present paper these results are extended to semiperf ect rings and generalizations of the three theorems are obtained. It is shown that a semiperfect ring with cyclic group of units is finite and is either commutative or is the direct sum of a commutative ring and the 2 x 2 upper triangular matrix ring over the field of two elements. Let R be semiperfect with an abelian group of units. It is shown that R is finite if either the group of units is finite or the group of units is finitely generated and the Jacobson radical is nil. The proofs of all these results depend on our main theorem: The structure of a semiperfect ring R with an abelian group of units is described completely up to the structure of commutative local rings. (That is commutative rings with a unique maximal ideal.) The groups of units of these local rings are shown to be direct factors of the group of units of R.

Two questions on semi-clean group rings

Let S be a cancellative monoid with quotient group of torsion-free rank a.We show that the monoid ring R[S] is a Hilbert ring if and only if the polynomial ring R[{ X, },s/] is a Hilbert ring, where |/| = a.Assume that R is a commutative unitary ring and G is an abelian group.The first research problem listed in [K, Chapter 7] is that of determining equivalent conditions in order that the group ring R[G] should be a Hilbert ring.Matsuda has considered this question in [Ml, 6].His results show that for G finitely generated, R[G] and R are simultaneously Hilbert rings; if G is not finitely generated, then R[G] a Hilbert ring implies that R is a Hilbert ring, but the converse fails.In [M2, 5] Matsuda considers briefly the corresponding Hilbert-ring-characterization problem for a monoid ring R[S], where S is torsion-free and cancellative.Let a be the torsion-free rank of G and let {Xj)lsl be a set of indeterminates over R of cardinality a.In Corollary 1 we show that R[G] and P[{^,},e/] are simultaneously Hilbert rings.While the problem of determining equivalent conditions for P[{Ar,}/e/] to be a Hilbert ring has not been completely resolved, it has been worked on extensively [Kr, Go, L, Gl, H], and a significant body of positive results exists concerning this problem.The proof of Theorem 1 suggests the following conjecture: if S is a cancellative commutative monoid with quotient group G, then R[S] is a Hilbert ring if and only if R[G] is a Hilbert ring; this result is established in Theorem 2.All monoids considered are assumed to be commutative, and rings are assumed to be commutative and unitary.The statement of Theorem 1 uses the following terminology.For a cardinal number a, an extension ring T of R is said to be a-generated over R if T is generated over R by a set of cardinality at most a.Theorem 1. Assume that X = {XA is a set of indeterminates of cardinality a over the ring R. Denote by Za the direct sum of a copies of the additive group Z of integers.The following conditions are equivalent.(1) R[X] is not a Hilbert ring.(2) There exists a prime ideal P of R such that R/P admits an a-generated extension ring that is a G-domain, but not afield.(3) The group ring R[Za] is not a Hilbert ring._

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.

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.

In this paper we give a detailed exposition of some of the results announced in [18]. The primary motivation for this work is Witt's observation [31, Satz 7] that if F is a field of characteristic #72, his ring 1V(F) of classes of anisotropic quadratic forms may be written as Z[G]/K where G is an abelian group of exponent two (actually G =F*/F*2), Z[G] is the integral group ring of G, and K is an ideal of Z[G] generated by elements of the form gl + g2-g3g4 and 1 + g5 with gi in G. Of course these elements can be described more explicitly, but for our purposes the only information we need about K is that any homomorphism of Z[G] -> Z sends K to 0 or to an ideal of the form 2nZ. In this introduction we shall call any ideal of Z[G] with this property admissible. In [26], Pfister proved certain structure theorems for IV(F) usilg his theory of multiplicative forms. His proofs were simplified in [11, 22, 23, 29, 30]. Harrison [11] and Leicht and Lorenz [22] gave important complements to Pfister's results concerning the ideal theory of WV(F). The main goal of our paper is to understand and generalize these structure theorems using standard techniques of commutative algebra. In [28] and [3], Scharlau and Belskii have introduced and studied Witt and Witt-Grothendieck rings for profinite groups. Their definitions generalize the usual notions of Wit.t rings, WV(F), and Witt-Grothendieck rings, K (F), of quadratic forms over fields of characteristic #72. These rings also have the form Z[G]/K with G an abelian group of exponent two and K an admissible ideal of Z[G]. Here we are mainly iinterested in another generalization of 1V(F) and K(F), namely the Witt rings, IV(C, J), and the Witt-Grothendieck rings, K(C,J) of classes of hermitian forms over a connected commutative semilocal ring C with involution J. Since J may be the identity these include the Witt and Witt-Grothendieck rings of classes of symmetric bilinear forms

This thesis consists of two independent chapters. The first chapter concerns the Smith Normal Form (SNF) over the integers ℤ of integral matrices. We consider the SNF of a matrix A to be the ratio of two ℤ-modules -- a finitely generated abelian group; this is called the Smith group of A. The Smith group provides a unified setting to present both new and old results. The new results concern the relationship between the eigenvalues of an integral matrix and its SNF. In particular, the multiplicities of integer eigenvalues are shown to relate to the multiplicities in the type of the Smith group. Bounds are also given for the exponent of the Smith group. In some cases, these are best possible. The old results discussed are the interlacing of the SNF in the case of augmented matrices and the symmetries of the SNF for certain combinatorial matrices. The latter results are extended to rectangular matrices. Numerous examples are given throughout, along with many conjectures based on computation. The second chapter generalizes the work of Pless, et al. on duadic codes and Q-codes. We take abelian group codes to be ideals in the group ring F[G], where G is a finite abelian group of odd order n and F is a finite field with characteristic relatively prime to n. We define generalized Q-codes from a pair of idempotents of F[G] and an automorphism of G which together obey two simple equations. These codes are (n, (n+1)/2) and (n, (n-1)/2) linear codes. We show that all of the properties of duadic and Q-codes generalize. In particular, we extend the results on the relationship of these codes to projective planes with regular automorphism group G. When F has characteristic 2, we give simple numerical conditions on G and F which determine when generalized Q-codes exist. We also give some techniques for constructing these codes.

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.

The representation of a ring as the ring of all the automorphisms of a primary abelian operator group expresses significant inner properties of this ring, since there exists essentially at most one such representation of a given ring. Thus it is the object of this investigation to expose invariant qualities of these automorphism rings. They are found to be pecularities of their ideal theory (Chapter III); and from these we select a complete set of postulates for the class of all the automorphism rings of primary abelian operator groups (Sections 12 and 13); thus proving incidentally that the structure of these rings is completely determined by their ideal theory (Section 14). The properties of the ideals in these rings reflect so completely the structure of the underlying operator group that we are able to prove-provided the rank of this group is at least three-the essential identity of the group of a-utomorphisms of the ring and of the group of projectivities (= biunivoque and monotone increasing transformations) of the system of admissible subgroups of the underlying operator group (Chapter II). Two [extreme] special cases may serve as an-i illustration for these theorems. Every projective geometry of finite dimension not less than three may be represented as the set of admissible subgroups of a suitable primary abelian operator group; and our characterization of the automorphisms of the automophism ring specializes in this case to the (well known) theorem that each automorphism of the automorphism ring (= ring of square matrices with coefficients from a suitable (not necessarily commutative) field) is induced by a so-called seini-linear transformation. If secondly G is an ordinary primary abelian group with the property that the least common multiple of the orders of the elements in G is a prime power p' and that G contains at least three independent elements of maximum order, then our theorem states that every automorphism of the automorphism ring is an inner auomorphism.

Let B be any nite commutative ring with identity. In this case, B(X; 1 p kZ0) B(X; 1 p 2Z0) B(X; 1 p Z0), where p is a prime number and k 1, is the descending chain of commutative semigroup rings. All these semigroup rings are containing the polynomial ring B(X;Z0). In this paper, we introduced a construction technique of cyclic codes through the semi- group ring B(X; 1 p kZ0) instead of a polynomial ring.

Units in regular abelian p-group rings

We explain a new relationship between formal group laws and ring spectra in\nstable homotopy theory. We study a ring spectrum denoted DB which depends on a\ncommutative ring B and is closely related to the topological Andre-Quillen\nhomology of B. We present an explicit construction which to every 1-dimensional\nand commutative formal group law F over B associates a morphism of ring spectra\nF_*: HZ --> DB from the Eilenberg-MacLane ring spectrum of the integers. We\nshow that formal group laws account for all such ring spectrum maps, and we\nidentify the space of ring spectrum maps between HZ and DB. That description\ninvolves formal group law data and the homotopy units of the ring spectrum DB.\n

Let $KG$ be the group ring of a group $G$ over a commutative ring $K$ with unity. The rings $KG$ are described for which $xx^\sigma =x^\sigma x$ for all $x=\sum _{g\in G}\alpha _gg\in KG$, where $x\mapsto x^\sigma =~\sum _{g\in G}\alpha _gf(g)\sigma (g)$ is an involution of $KG$; here $f: G\to U(K)$ is a homomorphism and $\sigma$ is an antiautomorphism of order two of $G$.