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## Projective S-acts and Exact Functors

### Algebra Colloquium (2000-03-01) 7: 113-120 , March 01, 2000

Let *S* be a semigroup. In this paper, projective *S*-acts and exact sequences in *S*-Act are studied. It is shown that, for a unitary *S*-act *P*, the functor Hom(*P*, –) is exact if and only if *P* ≅ *Se* for some idempotent *e* ɛ *S*.

## Subgroups Determined by Certain Products of Augmentation Ideals

### Algebra Colloquium (2000-03-01) 7: 1-4 , March 01, 2000

Let *G* be a group, *ZG* the integral group ring of *G*, and *I(G)* its augmentation ideal. Let *H* be a subgroup of *G*. It is proved that the subgroup of *G* determined by the product *I(H)I(G)I(H)* equals γ3(*H*), i.e., the third term in the lower central series of *H*. Also, the subgroup determined by *I(H)I(G)I*^{n}*(H)* (resp., *I*^{n}*(H)I(G)I(H)*) for *n* > 1 equals *D*_{n+2}(*H*), the (*n* + 2)th dimension subgroup of *H*.

## Simple Conformal Superalgebras of Finite Growth

### Algebra Colloquium (2000-04-01) 7: 205-240 , April 01, 2000

In this paper, we construct six families of infinite simple conformal superalgebras of finite growth based on our earlier work on constructing vertex operator superalgebras from graded assocaitive algebras. Three subfamilies of these conformal superalgebras are generated by simple Jordan algebras of types A, B, and C in a certain sense.

## Degree of Irrationality of Hyperelliptic Surfaces

### Algebra Colloquium (2000-08-01) 7: 319-328 , August 01, 2000

The degree of irrationality, which is a new birational invariant of algebraic varieties, has been introduced before. But the value has been determined only for a few varieties. In this paper, we show that the value for a hyperelliptic surface is 2 or 3 except one undetermined case. Especially, we give a characterization of some hyperelliptic surfaces by their function fields.

## A Singular Equation of Length Four Over Groups

### Algebra Colloquium (2000-08-01) 7: 247-274 , August 01, 2000

Let *G* be a group and *t* an unknown. In this paper we prove that the equation *atbtct*^{−1}*dt*^{−1} = 1 (*a,b,c,d* ɛ *G*, *a*^{2} ≠ 1, *c*^{2} ≠ 1, *bd* ≠ 1) has a solution over *G*. This forms part of a program to investigate precisely when an equation, whose associated star graph contains no admissible paths of length less than 3, fails to have a solution over *G*.

## Finitely Σ-CS Property of Excellent Extensions of Rings

### Algebra Colloquium (2003-06-01) 10: 17-21 , June 01, 2003

For a right excellent extension *S* of a ring *R*, it is proved that *R* is right, finitely Σ-CS if and only if *S* is the same. As an application of this result, a number of examples of group rings which are finitely Σ-CS are presented. This generalizes a result of Jain, et al. [5], where it was shown that *F*[*D*_{∞}] is CS when *F* is a field of characteristic ≠ 2. It is also proved that if *R* is a commutative domain with 2^{−1} ∈ *R* and C_{2} is the cyclic group of order 2, then *R*[*C*_{2}] is a CS-ring.

## Lattices of Radicals of Involution Rings

### Algebra Colloquium (2000-03-01) 7: 17-26 , March 01, 2000

A lattice-theoretic approach to the radical theory of rings was initiated by Snider. In the current paper, we extend this approach to the radical theory of involution rings. We show that the classes of hereditary radicals, radicals satisfying ADS and invariant radicals form complete sublattices of the lattice of all radicals of involution rings. We show that certain sublattices are isomorphic to sublattices of the lattice of radicals of rings. We characterize the atoms of certain lattices of radicals of involution rings.

## Derivations with Invertible or Nilpotent Values on a Multilinear Polynomial

### Algebra Colloquium (2000-03-01) 7: 93-98 , March 01, 2000

Let *R* be a prime ring with no non-zero nil one-sided ideals, *d* a nonzero derivation on *R*, and *f(X*_{1},...,*X*_{t}) a multilinear polynomial not central-valued on *R*. Suppose *d(f(x*_{1},...,*x*_{t})) is either invertible or nilpotent for all *x*_{1},...,*x*_{t} in some non-zero ideal of *R*. Then it is proved that *R* is either a division ring or the ring of 2 × 2 matrices over a division ring. This theorem is a simultaneous generalization of a number of results proved earlier.

## On a Problem of Karpilovsky

### Algebra Colloquium (2003-06-01) 10: 11-16 , June 01, 2003

Let *G* be a finite elementary group. Let Δ^{n} (*G*) denote the *n*th power of the augmentation ideal Δ(*G*) of the integral group ring Δ*G*. In this paper, we give an explicit basis of the quotient group *Q*_{n}(*G*) = Δ^{n}(*G*)/Δ^{n+1} (*G*) and compute the order of *Q*^{n} (*G*).

## Automorphisms of Association Schemes of Quadratic Forms over a Finite Field of Characteristic Two

### Algebra Colloquium (2003-06-01) 10: 63-74 , June 01, 2003

Let *X*_{n} denote the set of quadratic forms in *n* variables over a finite field *F*_{q} of characteristic 2, and *X*_{n} the association scheme on *X*_{n} by defining the relations with respect to the type of quadratic forms. We prove that every automorphism of *X*_{n} is of the form *X* ↦ *P*^{t}*X*^{σ}*P* + *Y* for all *X* ∈ *X*_{n}, where *P* ∈ *GL*_{n}(*F*_{q}), σ is an automorphism of *F*_{q}, and *Y* ∈ *X*_{n}.