Arithmetic of Semigroup Semirings

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.

The mini workshop "Arithmetic of group rings" was attended by 16 participants from Belgium, Brazil, Canada, Germany, Hungary, Israel, Italy, Romania and Spain. The expertise was a good mixture between senior and young researchers. It was a very stimulating experience and thesize of the group allowed excellent discussions amongst all participants. Very fruitful were the problem sessions, resulting in the problems listed at the end of this report. The main highlights of the conference were: The complete calculation of the projective Schur subgroup of the Brauer group by Aljadeff and del Rio. Hertweck's solution of the first Zassenhaus conjecture for finite metacyclic groups. the description of special subgroups of the unit group of integral group rings, such as the hypercentre and the finite conjugacy centre, and the relation with respect to the normalizer of the trivial units. discussion of the present state of art via several survey talks presented and the problem sessions The group G determines its integral group ring \mathbb Z G and its group V(\mathbb Z G) of normalized units. Several talks addressed the interplay of the cohomological properties of these three objects. Further topics included twisted group rings, group rings over local rings, polynomial growth and identities, orders and semigroup rings, Lie structure, representation-theoretic and algorithmic methods.

Noetherian Semigroup Rings with Several Objects

Let S be a cancellative semigroup. This paper is motivated by the problem of finding a description of semigroup rings K[S] over a field K that are semiprime or prime. Results of this type are well-known in the case of a group ring K[G], cf. [8]. The description, as well as the proofs, involve the FC-centre of G defined as the subset of all elements with finitely many conjugates in G. In [4], [5] Krempa extended the FC-centre techniques to the case of an arbitrary cancellative semigroup S. He defined a subsemigroup Δ(S) of S which coincides with the FC-centre in the case of groups, and can be used to describe the centre and to study special elements of K[S]. His results were strengthened by the author in [7], where Δ(S) was also applied in the context of prime and semiprime algebras K[S]. However, Δ(S) itself is not sufficient to characterize semigroup rings of this type. We note that in [2], [3] Dauns developed a similar idea for a study of the centre of semigroup rings and certain of their generalizations.

Throughout this paper the ring R and the semigroup S are commutative with identity; moreover, it is assumed that S is cancellative, i.e., that S can be embedded in a group. The aim of this note is to determine necessary and sufficient conditions on R and S that the semigroup ring R[S] should be one of the following types of rings: principal ideal ring (PIR), ZPI-ring, Bezout, semihereditary or arithmetical. These results shed some light on the structure of semigroup rings and provide a source of examples of the rings listed above. They also play a key role in the determination of all commutative reduced arithmetical semigroup rings (without the cancellative hypothesis on S) which will appear in a forthcoming paper by Leo Chouinard and the authors [4].

The aim of this thesis is to investigate the circumstances under which group rings over fields have non-zero socle, i.e. contain minimal one-sided ideals. After an introductory chapter, we consider the special case o₊ a periodic abelian group and a non-modular field (that is, a field of characteristic prime to the orders of the elements of the group). This special case, and the background material contained in Chapter III, serve as preparation for our principal results, which concern, locally finite groups. We establish necessary and sufficient conditions on an arbitrary field K and a locally finite group 0 for the group ring KG to contain minimal one-sided ideals: the most important condition is that G should be a Cernikov group. We then examine the structure of KG when these conditions are satisfied. Vie show that KG has a finite series of ideal3 each factor of which i3 a direct sum of quasi-Frobenius rings, and characterize the socle of KG. Me also classify indecomposable KG-modules, and determine (for countable but not necessarily locally finite groups G) necessary and sufficient conditions for all indecomposable KG-modules to be irreducible. In the final chapter we consider non-locally-finite groups, conjecturing that group rings of such groups never contain minimal one-sided ideals. We establish the truth of this conjecture for several classes of groups, and also consider semiartinian group rings.

The notion of an almost Krull domain is extended to rings satisfying a polynomial identity. Some general structural results are obtained. We also prove that skew polynomial rings R [ X, σ] remain almost Krull if R is an almost Krull ring. Finally, we study when semigroup ring R[S] are almost Krull rings, in the case when the group of quotients of S has the ascending chain condition on cyclic subgroups. An example is included to show that the general (semi-) group ring case is much more difficult to deal with.

Typical primitive polynomials over integer residue rings

Let K[G] denote the group ring of G over the field K. In this note we characterize those group rings in which all left ideals are right ideals. Let R be a ring. We say that R is l.i.r.i. if every left ideal is a right ideal. A ring is l.a.r.i. if every left annihilator is a right ideal. Our notation follows that of [2]. The main results are THEOREM I. Let K be a field and let G be a nonabelian periodic group. Then if K[G] is l.a.r.i. one of the following occurs (i) Char K = 0 and G is a Hamiltonian group such that for each odd exponent, n, of G the quaternion algebra over the field K(4), where 4 is a primitive nth root of unity, is a division ring. (ii) Char K = 2 and K does not contain any primitive cube root of unity. Moreover G = Q x A, where Q is the quaternion group of order 8 and A is abelian in which each element has odd order and if n is an exponent for A, the least integer m > 1 satisfying 2m -1 (mod n) is odd. Conversely if K[G] satisfies either (i) or (ii), then K[G] is l.i.r.i. and, in particular, it is l.a.r.i. Observe that if char K > 2 and G is periodic, then K[GI is l.a.r.i. if and only if G is abelian. THEOREM II. Let K[G] denote the group ring over a nonabelian group. Then the following are equivalent (i) K[G] is l.i.r.i. (ii) G is locally finite and if a,f8 E K[GI with a,8 = 0, then /3a = 0. (iii) G is locally finite and K[G] is l.a.r.i. If we combine the above theorems we get necessary and sufficient conditions for K[GI to be l.i.r.i. By using the antiautomorphism of K[G] given by ,xeG k.x H* IeG kxxwe see that K[G] is l.i.r.i. (l.a.r.i.) if and only if K[G] is r.i.l.i. (r.a.l.i.). Received by the editors February 2, 1978. AMS (MOS) subject classifications (1970). Primary 16A26.

In this paper, we study the question of when the symmetric units in an integral group ring ZG form a multiplicative group. When G is periodic, necessary and sufficient conditions are given for this to occur.

It is proved that from the isomorphism of the integral group rings of two groups follows the isomorphism of the quotient-groups of these groups by the second commutator subgroup. Groups with certain conditions T1 and T2 are considered. From condition T1 it follows that the group algebra over an integral domain has no zero-divisors, and from T2 that the group of invertible elements of this algebra is trivial. Sufficient conditions are given, under which there are no zero-divisors in the group algebra of a pro-p-group over a field of p elements.

The purposes of the present paper are (1) to give a necessary and sufficient condition for the uniqueness of the separable idempotent for a separable group ring extension RG(R may be a non‐commutative ring), and (2) to give a full description of the set of separable idempotents for a quaternion ring extension RQ over a ring R, where Q are the usual quaternions i, j, k and multiplication and addition are defined as quaternion algebras over a field. We shall show that RG has a unique separable idempotent if and only if G is abelian, that there are more than one separable idempotents for a separable quaternion ring RQ, and that RQ is separable if and only if 2 is invertible in R.

Let $\Lambda$ be a ring endowed with an involution $a \mapsto \tilde{a}$. We say that two units $a$ and $b$ of $\Lambda$ fixed under the involution are congruent if there exists an element $u \in \Lambda^{\times}$ such that $a = ub\tilde{u}$. We denote by $\mathcal{H}(\Lambda)$ the set of congruence classes. In this paper we consider the case where $\Lambda$ is an order with involution in a semisimple algebra $A$ over a local field and study the question of whether the natural map $\mathcal{H}(\Lambda) \rightarrow \mathcal{H}(\Lambda)$ induced by inclusion is injective. We give sufficient conditions on the order $\Lambda$ for this map to be injective and give applications to hermitian forms over group rings.

Testing for delay and CMOS stuck-open faults requires two pattern tests and test sets are usually large. Built-in self-test (BIST) schemes are attractive for such comprehensive testing. The BIST test pattern generators (TPGs) for such testing should be designed to ensure high pattern-pair coverage. In this paper, necessary and sufficient conditions to ensure complete/maximal pattern-pair coverage for linear feedback shift register (LFSR) and cellular automata (CA) have been derived. The theory developed here identifies all LFSR/CA TPGs which maximize pattern-pair coverage under any given TPG size constraints. It is shown that LFSRs with primitive feedback polynomials with large number of terms are better for two-pattern testing. Also, CA are shown to be better TPGs than LFSRs for two-pattern testing. Results derived in this paper provide practical algorithms for the design of optimal TPGs for two-pattern testing. Experiments on some benchmark circuits indicate the TPGs designed using the procedures outlined in this paper provide much higher delay fault coverage than other TPGs. >

In this paper, we introduce the concepts of semi-Baer, semi-quasi Baer, semi-p.q. Baer and semi-p.p. modules as a generalization of Baer, quasi Baer, p.q. Baer and p.p. modules respectively. To define these concepts, we introduce concepts of multiplicative order of an element and a multiplicatively finite element in rings. Further, we characterize these concepts in modules over reduced rings. Also, it is proved that semi-Baer and semi-quasi Baer properties are preserved by polynomial extensions and power series extensions of modules. It is proved that for a ring R and a monoid G, if the semi group ring RG is semi-Baer (semi-quasi Baer) then so is R.