Projective Congruence in M3(Fq

(3 × 3) matrices are here classified up to the relation of projective congruence. This is then applied to obtain the classification up to isomorphism of a certain class of finite rings of characteristic p. These rings arise naturally in the recent determination of all rings of order p ⁿ (n ≤ 5) by B. Corbas and the author, and the work here completes that study.     

On Galois structure of the integers in elementary abelian extensions of local number fields

Let p be an odd prime number and k a finite extension of Qp. Let K/k be a totally ramified elementary abelian Kummer extension of degree p² with Galois group G. We determine the isomorphism class of the ring of integers in K as an oG-module under some assumptions. The obtained results imply there exist extensions whose rings are ZpG-isomorphic but not oG-isomorphic, where Zp is the ring of p-adic integers. Moreover we obtain conditions that the rings of integers are free over the associated orders and give extensions whose rings are not free.     

Negacyclic codes over Galois rings of characteristic 2 a

We investigate negacyclic codes over the Galois ring GR(2 a ,m) of length N = 2 k n,where n is odd and k 0.We first determine the structure of u-constacyclic codes of length n over the finite chain ring GR(2 a ,m)[u]/ u 2 k + 1 .Then using a ring isomorphism we obtain the structure of negacyclic codes over GR(2 a ,m) of length N = 2 k n (n odd) and explore the existence of self-dual negacyclic codes over GR(2 a ,m).A bound for the homogeneous distance of such negacyclic codes is also given.     

p-Rank and semi-stable reduction of curves

Let R be a discrete complete valuation ring, with field of fractions K, and with algebraically closed residue field k of characteristic p > 0. Let X be a germ of an R-curve at an ordinary double point. Consider a finite Galois covering f: Y → X, whose Galois group G is a p-group, such that Y is normal, and which is étale above X_k≔ x × _rk. Asume that Y has a semi-stable model :→ Y over R, and let y be a closed point of Y. If the inertia subgroup I(y) at y is cyclic of order pⁿ, we compute the p-rank of tf⁻¹ (y) by using a result of Raynaud. In particular, we prove that this p-rank is bounded by pⁿ −1.     

Submodule categories of wild representation type

Let Λ be a commutative local uniserial ring with radical factor field k. We consider the category S(Λ) of embeddings of all possible submodules of finitely generated Λ-modules. In case Λ=Z/〈pⁿ〉, where p is a prime, the problem of classifying the objects in S(Λ), up to isomorphism, has been posed by Garrett Birkhoff in 1934. In this paper we assume that Λ has Loewy length at least seven. We show that S(Λ) is controlled k-wild with a single control object I∈S(Λ). It follows that each finite dimensional k-algebra can be realized as a quotient End(X)/End(X)_I of the endomorphism ring of some object X∈S(Λ) modulo the ideal End(X)_I of all maps which factor through a finite direct sum of copies of I.     

Number of Zeros of Diagonal Polynomials over Finite Fields

Let F be a finite field with q=pf elements, where p is a prime. Let N be the number of solutions (x1,…,xn) of the equation c1xd11+···+cnxdnn=c over the finite fields, where d1∣q−1, ciϵF*(i=1, 2,…,n), and cϵF. In this paper, we prove that if b1 is the least integer such that b1≥∑ni=1 (f/ri) (Di, p−1)/(p−1), then q[b1/f]−1∣N, where ri is the least integer such that di∣pri−1, Didi=pri−1, the (Di, p−1) denotes the greatest common divisor of Di and p−1, [b1/f] denotes the integer part of b1/f. If di=d, then this result is an improvement of the theorem that pb∣N, where b is an integer less than n/d, obtained by J. Ax (1969, Amer. J. Math.86, 255–261) and D. Wan (1988, Proc. AMS103, 1049–1052), under a certain natural restriction on d and n.     

Determinants of matrices over commutative finite principal ideal rings

In this paper, the determinants of $n\times n$ matrices over commutative finite chain rings and over commutative finite principal ideal rings are studied. The number of $n\times n$ matrices over a commutative finite chain ring R of a fixed determinant a is determined for all $a\in R$ and positive integers n. Using the fact that every commutative finite principal ideal ring is a product of commutative finite chain rings, the number of $n\times n$ matrices of a fixed determinant over a commutative finite principal ideal ring is shown to be multiplicative, and hence, it can be determined. These results generalize the case of matrices over the ring of integers modulo m.     

Values of coefficients of cyclotomic polynomials

Let a(k,n) be the k-th coefficient of the n-th cyclotomic polynomials. In 1987, J. Suzuki proved that . In this paper, we improve this result and prove that for any prime p and any integer l≥1, we have

{a(k,p^ln)∣n,k∈N}=Z.     

Universal Deformation Rings of Modules for Algebras of Dihedral Type of Polynomial Growth

Let k be an algebraically closed field, and let Λ be an algebra of dihedral type of polynomial growth as classified by Erdmann and Skowroński. We describe all finitely generated Λ-modules V whose stable endomorphism rings are isomorphic to k and determine their universal deformation rings R(Λ, V). We prove that only three isomorphism types occur for R(Λ, V): k, k[[t]]/(t ²) and k[[t]].     

On LCD,self dual and isodual cyclic codes over finite chain rings

In this paper, LCD cyclic, self dual and isodual codes over finite chain rings are investigated. It was proven recently that a non-free LCD cyclic code does not exist over finite chain rings. Based on algebraic number theory, we introduce necessary and sufficient conditions for which all free cyclic codes over a finite chain ring are LCD. We have also obtained conditions on the existence of non trivial self dual cyclic codes of any length when the nilpotency index of the maximal ideal of a finite chain ring is even. Further, several constructions of isodual codes are given based on the factorization of the polynomial $x^n-1$ over a finite chain ring.     

The topology determined by the symbolic powers of primary ideals

Let q be a p-primary ideal in a Noetherian ring R. The main theorem characterizes when the q-adic and q-symbolic topologies on R are linearly equivalent; that is, when there n kexists an integer k≥0 such that q⁽ⁿ⁾ ?q^n-k for all n≥k. Using this, it is shown that when this holds for q, then it holds for several other primary ideals related to q (both in R and in certain other rings related to R) and that the powers of q(ⁿ) are symbolic powers for all n≥k (so p has primary ideals all of whose powers are symbolic powers).     

Global properties of tensor rank

The dependence of tensor rank on the underlying ring of scalars is considered. It is shown that the integers are, in a certain sense, the worst scalars. A ring of scalars can be improved by adjoining algebraic elements but not by adjoining indeterminates. The real closed fields are the best scalars among ordered rings, and the algebraically closed fields are best among all rings. Let B(R^m×n×p) be the maximum tensor rank of any m×n×p array of elements from the ring R. A generalization of Gaussian elimination shows that B(R^n×n×n)?$\text{3}\text{4}$n² for most useful rings R. For every R, B(R^m×n×p)?mnp/(m+n+p), and slightly stronger lower bounds are proven for R a field.     

Classification of reflexible regular embeddings and self-Petrie dual regular embeddings of complete bipartite graphs

In this paper, we classify the reflexible regular orientable embeddings and the self-Petrie dual regular orientable embeddings of complete bipartite graphs. The classification shows that for any natural number n, say (p₁,p₂,…,p_k are distinct odd primes and a_i>0 for each i?1), there are t distinct reflexible regular embeddings of the complete bipartite graph K_n,n up to isomorphism, where t=1 if a=0, t=2^k if a=1, t=2^k+1 if a=2, and t=3·2^k+1 if a?3. And, there are s distinct self-Petrie dual regular embeddings of K_n,n up to isomorphism, where s=1 if a=0, s=2^k if a=1, s=2^k+1 if a=2, and s=2^k+2 if a?3.     

The K-theory of fields in characteristic p

We show that for a field k of characteristic p, H ⁱ (k,ℤ(n)) is uniquely p-divisible for i≠n (we use higher Chow groups as our definition of motivic cohomology). This implies that the natural map K _n ^M (k)?K _n (k) from Milnor K-theory to Quillen K-theory is an isomorphism up to uniquely p-divisible groups, and that K _n ^M (k) and K _n (k) are p-torsion free. As a consequence, one can calculate the K-theory mod p of smooth varieties over perfect fields of characteristic p in terms of cohomology of logarithmic de Rham Witt sheaves, for example K _n (X,ℤ/p ^r )=0 for n>dimX. Another consequence is Gersten’s conjecture with finite coefficients for smooth varieties over discrete valuation rings with residue characteristic p. As the last consequence, Bloch’s cycle complexes localized at p satisfy all Beilinson-Lichtenbaum-Milne axioms for motivic complexes, except possibly the vanishing conjecture. Oblatum 21-I-1998 & 26-VII-1999 / Published online: 18 October 1999     

The Ring of Modules with Endo-permutation Source

In this paper we consider certain subalgebras of the Green algebra (representation algebra) of a finite group G. One such algebra is spanned by the isomorphism classes of all indecomposable modules whose source is an endo-permutation module. This algebra can be embedded into a finite direct product of Laurent polynomial rings in finitely many variables over a field. Another such algebra is spanned by the isomorphism classes of all irreducibly generated modules. When G is p-solvable then this algebra is finite-dimensional and split semisimple.R. Boltje was supported by the NSF, DMS-0200592 and 0128969. B. Külshammer was supported by the DAAD.     

Lattice-Ordered Matrix Rings Over Totally Ordered Rings

For an n ×n matrix algebra over a totally ordered integral domain, necessary and sufficient conditions are derived such that the entrywise lattice order on it is the only lattice order (up to an isomorphism) to make it into a lattice-ordered algebra in which the identity matrix is positive. The conditions are then applied to particular integral domains. In the second part of the paper we consider n ×n matrix rings containing a positive n-cycle over totally ordered rings. Finally a characterization of lattice-ordered matrix ring with the entrywise lattice order is given.     

Finite 2-groups whose length of chain of nonnormal subgroups is at most 2

Assume that G is a finite non-Dedekind p-group. D. S. Passman introduced the following concept: we say that H₁ < H₂ < ? < H_k is a chain of nonnormal subgroups of G if each H_i ? G and if |H_i : H_i?1| = p for i = 2, 3,…, k. k is called the length of the chain. chn(G) denotes the maximum of the lengths of the chains of nonnormal subgroups of G. In this paper, finite 2-groups G with chn(G) ? 2 are completely classified up to isomorphism.     

Endomorphism Algebras of Some Modules for Schur Algebras and Representation Dimension

We consider the representation dimension, for fixed n ≥ 2, of ordinary and quantised Schur algebras S(n, r) over a field k. For k of positive characteristic p we give a lower bound valid for all p. We also give an upper bound in the quantum case, when k has characteristic 0.     

Automorphism groups of generalized triangular matrix rings

We call a ring strongly indecomposable if it cannot be represented as a non-trivial (i.e. M≠0) generalized triangular matrix ring , for some rings R and S and some R-S-bimodule _RM_S. Examples of such rings include rings with only the trivial idempotents 0 and 1, as well as endomorphism rings of vector spaces, or more generally, semiprime indecomposable rings. We show that if R and S are strongly indecomposable rings, then the triangulation of the non-trivial generalized triangular matrix ring is unique up to isomorphism; to be more precise, if is an isomorphism, then there are isomorphisms ρ:R→R^′ and ψ:S→S^′ such that χ:=φ_∣M:M→M^′ is an R-S-bimodule isomorphism relative to ρ and ψ. In particular, this result describes the automorphism groups of such upper triangular matrix rings      

Factorisation of Lie resolvents

Let G be a group, F a field of prime characteristic p, and V a finite-dimensional FG-module. For each positive integer r, the rth homogeneous component of the free Lie algebra on V is an FG-module called the rth Lie power of V. Lie powers are determined, up to isomorphism, by certain functions Φ^r on the Green ring of FG, called ‘Lie resolvents’. Our main result is the factorisation Φ^pmk=Φ^pm°Φ^k whenever k is not divisible by p. This may be interpreted as a reduction to the key case of p-power degree.