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In this paper we give a complete classification of strongly square-free reflective Z -lattices of signature ( 5 , 1 ) . This is done by reducing the classification of Lorentzian lattices to those of positive-definite lattices. The classification of totally-reflective genera breaks up into two parts. The first part consists of classifying the square free, totally-reflective, primitive genera by calculating strong bounds on the prime factors of the determinant of positive-definite quadratic forms (lattices) with this property. We achieve these bounds by combining the Minkowski–Siegel mass formula with the combinatorial classification of reflective lattices accomplished by Scharlau & Blaschke. In a second part, we use a lattice transformation that goes back to Watson, to generate all totally-reflective, primitive genera when starting from the square free case.

We define two different simplicial complexes, the common divisor simplicial complex and the prime divisor simplicial complex, from a set of integers, and explore their similarities. We will define a map between the two simplicial complexes, and use this map to show that for any set of integers, the fundamental groups of the resulting simplicial complexes are isomorphic.

In this note, it is shown that a finite group G is solvable if for each odd prime divisor p of | G | , | Irr ( B 0 ( G ) 2 ) ∩ Irr ( B 0 ( G ) p ) | ≤ 2 , where Irr ( B 0 ( G ) p ) is the set of complex irreducible characters of the principal p-block B 0 ( G ) p of G. Also, the structure of such groups is investigated. Examples show that the bound 2 is best possible.

Let G be a finite group and let r be a prime divisor of the order of G. We prove that if r ≥ 5 and G has the E { r , t } -property for all t ∈ π ( G ) \ { r } , then G is r-solvable. A group G is said to have the E π -property if G possesses a Hall π-subgroup.

Let G be a (topological) group. For 2 ⩽ d ∈ N , denote by μ d ( G ) the largest m for which there exists an m -tuple of elements of G such that any of its d entries generate G (topologically). We obtain a lower bound for μ d ( G ) in the case when G is a prosolvable group. Our result implies in particular that if G is d -generated then the difference μ d ( G ) − d tends to infinity when the smallest prime divisor of the order of G tends to infinity. One of the aim of the paper is to draw the attention to an intriguing question in linear algebra whose solution would allow to improve our bounds and determine the precise value for μ d ( G ) in several relevant cases, for example when d = 2 and G is a prosolvable group.

We study prime algebras of quadratic growth. Our first result is that if $A$ is a prime monomial algebra of quadratic growth then $A$ has finitely many prime ideals $P$ such that $A/P$ has GK dimension one. This shows that prime monomial algebras of quadratic growth have bounded matrix images. We next show that a prime graded algebra of quadratic growth has the property that the intersection of the nonzero prime ideals $P$ such that $A/P$ has GK dimension 2 is non-empty, provided there is at least one such ideal. From this we conclude that a prime monomial algebra of quadratic growth is either primitive or has nonzero locally nilpotent Jacobson radical. Finally, we show that there exists a prime monomial algebra $A$ of GK dimension two with unbounded matrix images and thus the quadratic growth hypothesis is necessary to conclude that there are only finitely many prime ideals such that $A/P$ has GK dimension 1., 23 pages

Gluck proved that any finite group G has an abelian subgroup A such that | G : A | is bounded by a polynomial function of the largest degree of the complex irreducible characters of G . This improved on a previous bound of Isaacs and Passman. In this paper, we present a variation of this result that looks at the number of prime factors. All these results, in turn, may be seen as variations on the classical theorem of Jordan on linear groups.

Let S = { x 1 , … , x n } be a set of n distinct positive integers. Let f be an arithmetical function. Let [ f ( x i , x j ) ] denote the n × n matrix having f evaluated at the greatest common divisor ( x i , x j ) of x i and x j as its i , j -entry and ( f [ x i , x j ] ) denote the n × n matrix having f evaluated at the least common multiple [ x i , x j ] of x i and x j as its i , j -entry. The set S is said to be gcd-closed if ( x i , x j ) ∈ S for all 1 ⩽ i , j ⩽ n . For an integer x, let ν ( x ) denote the number of distinct prime factors of x. In this paper, by using the concept of greatest-type divisor introduced by S. Hong in [Adv. Math. (China) 25 (1996) 566–568; J. Algebra 218 (1999) 216–228], we obtain a new reduced formula for det f [ ( x i , x j ) ] if S is gcd-closed. Then we show that if S = { x 1 , … , x n } is a gcd-closed set satisfying max x ∈ S { ν ( x ) } ⩽ 2 , and if f is a strictly increasing (respectively decreasing) completely multiplicative function, or if f is a strictly decreasing (respectively increasing) completely multiplicative function satisfying 0 f ( p ) ⩽ 1 p (respectively f ( p ) ⩾ p ) for any prime p, then the matrix [ f ( x i , x j ) ] (respectively ( f [ x i , x j ] ) ) defined on S is nonsingular. As a corollary, we show the following interesting result: The LCM matrix ( [ x i , x j ] ) defined on a gcd-closed set is nonsingular if max x ∈ S { ν ( x ) } ⩽ 2 .

Let H be a finite group, and let θ be an automorphism of H whose order divides n. Hall proved that the number of elements x of H that satisfy the relation xxθxθ2⋯xθn−1=1 is a multiple of gcd(n,|H|). For a prime factor p of gcd(n,|H|), if such a number is not a multiple of gcd(pn,|H|), then a Sylow p-subgroup of H is exceptional.

For every positive integer n and every finite set S of prime numbers, we construct An-extensions of Q unramified at all primes in S∪{∞}; moreover, these extensions are obtained as splitting fields of totally real monic polynomials in Z [X] of degree n whose discriminant is not divisible by any prime number p in S. As a corollary, we obtain that there exist infinitely many linearly disjoint tamely ramified An-extensions of Q .

For a submonoid S of a torsion-free abelian-by-finite group, we describe the height-one prime ideals of the semigroup algebra K [ S ]. As an application we investigate when such algebras are unique factorization rings.

A settiles the integersif and only if the integers can be written as a disjoint union of translates of that set. We consider the problem of finding necessary and sufficient conditions for a finite set to tile the integers. For sets of prime power size, it was solved by D. Newman (1977,J. Number Theory9, 107–111). We solve it for sets of size having at most two prime factors. The conditions are always sufficient, but it is unknown whether they are necessary for all finite sets.

Suppose that the finite abelian group G is a direct product of the subsets A1,..., An such that for each i, there is a subgroup Hi of G with |Ai| = |Hi| ≤ |Ai ∩ Hi| + p − 2, where p is the least prime factor of |G|. A. D. Sands proved that then Ai = Hi for some i. We prove that the same conclusion holds if |Hi| = |Ai| ≤ |Ai ∩ Hi| + pi − 2 for each i, where pi is the least prime factor of |Ai|. As generalizations of earlier results we prove two similar results.

The purpose of the article is to prove that for any regular right self-injective ring R the center of any factor ring of R is a factor ring of the center of R. It is also proved that the extended centroid of any prime factor ring of R is equal to center of this factor ring.

Gluck [2] and others have investigated the relationship between p(G), the set of primes dividing the degrees of the irreducible complex characters of G, and o(G), the greatest number of primes dividing the degree of a single irreducible character of G. when G is a finite solvable group. For such groups Gluck [2] has shown that 1 p(G)/ ,< (g(G))‘+ 100(G). In this paper we examine the relationship between these two quantities for G a finite nonabelian simple group. In order to state our results we need to introduce some notation. If G is a finite group and S is a subset of Irr(Gj, the set of irreducible characters of G, we will say S is a covering set of G if for every prime divisor p of / G 1 there is a character x in S such that p divides x( 1). If G has a covering set we will define the covering number of G, which we will denote by en(G), as the least number of elements in a covering set of G. Work of Michler [6, Theorem 3.31 has as a consequence that if G is a nonabelian simple group then G has a covering set; in other words, p(G) = n(G), where n(G) is the set of prime divisors of the order of G. For such groups then en(G) ,( 1 r(G)1 However, more is true.

Let I a denote the integral closure of an ideal I in a Noetherian ring A . Then it is shown that sup a ∞ I , the maximum number of elements which are ( I n ) a -independent as n → ∞, exists and is equal to min{height (I(A P ) ∗ + z) z : P is a prime divisor of ( I n ) a for all large n and z is a minimal prime ideal in the completion (A p ) ∗ of A p } . From this it follows that A is locally quasi-unmixed if and only if sup a ∞ I = height I for all ideals I in A .

Let I be a regular proper ideal in a Noetherian ring R. We prove that there exists a simple free integral extension ring A of R such that the ideal IA has a Rees-good basis; that is, a basis c1,…,cg such that ciW=IW for i=1,…,g and for all Rees valuation rings W of IA. Moreover, A may be constructed so that: (i) IA and I have the same Rees integers (with possibly different cardinalities), and (ii) AP is unramified over RP∩R for each asymptotic prime divisor P of IA. Indeed, if H is a regular ideal in R such that each asymptotic prime divisor of H is contained in an asymptotic prime divisor of I, then (ii) holds for HA. If Card(ReesH)⩽Card(ReesI), we prove that (i) also holds for HA and H. If I=(b1,…,bg)R and b1,…,bg is an asymptotic sequence, we prove that b1,…,bg is a Rees-good basis of I.

Let R be a Noetherian ring, and let UC Spec R. Let T, be the transform (~1 u is in the total quotient ring of R, and (R: U) & P for all PE U>. Let B(R) = {P E Spec R I grade P, = 1 }, and let 8(R) = {P E Spec R 1 the completion (R,)* has a depth 1 prime divisor of zero). The main result of the first section says that Tci is a finite R-module if and only if every regular prime in &P(R) is contained in some prime in U, and with linitely many exceptions, every regular prime in g(R) is contained in some prime in U. We then apply this result to various well-known transforms of R. A transform which recently has drawn interest, is R’ = T, for U = d?(R). It was known that R” is the union of all ideal transforms of R which happen to be finite R-modules, but it was not known when R’ itself is finite. Here, we show that R” is a finite R-module if and only if the set A(K)= (PE@(R)I no prime in d(R) contains P) is finite. The second section of the paper studies when A(R), and the larger set JL-( R) = Y(R) b(R), are finite. We show that this occurs quite frequently. A major result is that if R E -4 c B are Noetherian rings with B finitely generated over R, and if R is semilocal, then M(A) is finite, and so A’ is a finite A-module. The third section asks some questions, and makes related comments.

Let R be a Noetherian ring, F:=Rr and M⊆F a submodule of rank r. Let A∗¯(M) denote the stable value of Ass(Fn/Mn¯), for n large, where Fn is the nth symmetric power of Fn and Mn is the image of the nth symmetric power of M in Fn. We provide a number of characterizations for a prime ideal to belong to A∗¯(M). We also show that A∗¯(M)⊆A∗(M), where A∗(M) denotes the stable value of Ass(Fn/Mn).

Resume Consider a fieldk, some nonzero elementqofkwhich is not a root of unity, and some nonnegative integern. K. R. Goodearl and E. S. Letzter have proved (1991, Prime factor algebras of the coordinate ring of quantum matrices, preprint) that any prime factor algebra of the coordinate ringOq( m n(k)) of quantumn×nmatrices overkis a domain. In that paper it is proved that the division ring of fractions of such a domain is always isomorphic to the division ring of fractions of the coordinate ring of some quantum space of dimension at mostn2over some fieldKwhich is an extension ofk. A similar result is also proved for quantum Weyl algebras.

Let G be any group and a1G1,…,akGk (k>1) be left cosets in G. In 1974 Herzog and Schönheim conjectured that if A={aiGi}i=1k is a partition of G then the (finite) indices n1=[G:G1], …, nk=[G:Gk] cannot be pairwise distinct. In this paper we show that if A covers all the elements of G the same number of times and G1,…,Gk are subnormal subgroups of G not all equal to G, then M=max1⩽j⩽k|{1⩽i⩽k:ni=nj}| is not less than the smallest prime divisor of n1⋯nk; moreover, min1⩽i⩽klogni=O(Mlog2M) where the O-constant is absolute.

Let J= (b0, b1, …, bh) A be an ideal in a Noetherian ring A such that b0 is nonnilpotent and let B = A[X1, …, Xh]. Then a complete description of the asymptotic, essential, and quintessential prime divisors of K = (b0X1 − b1, …, b0Xh − bh) B is given, and a more detailed description of these prime divisors of K and of Ker( B → A[ b 1 b 0 , …, b h b 0 ] ) is given when b0, b1,…, bh are an asymptotic, essential, or A-sequence. Also, it is shown that there exists a unique essential prime divisor of K if and only if b0, b1, …, bh are an essential sequence and A has exactly one prime divisor of zero.

Let I be an ideal in a local ring (R, M) and let s(I) be the asymptotic cograde of I (= the maximum length of an asymptotic sequence over I). Rees has shown that s(I) ⩽ altitude R − l(I), where l(I) is the analytic spread of I. We prove several more bounds on s(I): (i) s(I) ⩽ min{little depth P; P is an asymptotic prime divisor of I; (ii) s(I) ⩽ grade ∗ M I (= the maximum length of an asymptotic sequence contained in M I ) ; (iii) s(I) ⩽ grade ∗ M- grade ∗ I ; (iv) grade ∗ M − l(I) ⩽ s(I) ; and. (v) if R is quasi-unmixed, then grade R I n ⩽ s(I) for all large n. Also, it is shown that the asymptotic prime divisors of (I, b1,…, bs) R contain those of I, when b1,…, bs are an asymptotic sequence over I, and that the residue classes modulo I of such a sequence are an asymptotic sequence in R I . Finally, several sufficient conditions for l(I) to be equal to l(Ip) for some asymptotic prime divisor P of I are given.

We study additive isomorphisms of prime rings preserving a multilinear polynomial of degree ≥2. Our main theorem generalizes a number of results obtained for Jordan, n-Jordan, Lie, and Lie triple isomorphisms of prime rings.

We characterize the fixed divisor of a polynomial [Formula: see text] in [Formula: see text] by looking at the contraction of the powers of the maximal ideals of the overring [Formula: see text] containing [Formula: see text]. Given a prime p and a positive integer n, we also obtain a complete description of the ideal of polynomials in [Formula: see text] whose fixed divisor is divisible by [Formula: see text] in terms of its primary components.