Algebra Colloquium

In this paper, we find the structure of finite groups G=AB with subgroups A ≅ L₂ (7) or L₂ (8) and B ≅ S_n (n ≥ 5).

In this paper, among other results, we prove that a clone C with five essentially binary operations is minimal if and only if C is a clone of a non-trivial affine space over GF(7). This result is a product of systematic investigation of varieties of idempotent commutative groupoids. It is probably the most spectacular result of this investigation, but many other results formulated and proved throughout the paper may turn out more useful in further study in this area.

New examples of metaideals in commutative rings are constructed. It is proved that metaideals of a commutative ring form a sublattice of the lattice of all subrings, and for any subring A of a commutative ring P, there exists the largest subring Mid_P (A) (called metaidealizer) in which A is a metaideal. Metaidealizers in several cases are described.

We study symmetric powers of classes of symmetric bilinear forms in the Witt–Grothendieck ring of a field of characteristic not equal to 2, and derive their basic properties and compute their classical invariants. We relate these to earlier results on exterior powers of such forms.

In this paper, we show that CL_n, the chain with n elements, is efficient and that the direct product CL_m × CL_n is inefficient. Moreover, we embed any finitely presented semigroup S into an inefficient semigroup, namely, the semigroup S ⋃ SL_n, where SL_n is the free semilattice of rank n.

Quaternion algebras over curves of genus one without rational points over perfect fields of characteristic not two are classified. As a consequence, all quaternion algebras over the function field of such a curve, which are unramified everywhere, are classified.

The set of all endomorphisms of an algebraic structure with composition of functions as operation is a rich source of semigroups, which has only rarely been dipped into (see [2]). Here we make a start by considering endomorphisms of Clifford semigroups, relating them to the homomorphisms and endomorphisms of the underlying groups.

For near-ring ideal mappings ρ₁ and ρ₂, we investigate radical theoretical properties of and the relationship among the class pairs (ρ₁: ρ₂), and (ℛ_ρ2: ℛ_ρ1). Conditions on ρ₁ and ρ₂ are given for a general class pair to form a radical class of various types. These types include the Plotkin and KA-radical varieties. A number of examples are shown to motivate the suitability of the theory of Hoehnke-radicals over KA-radicals when radical pairs of near-rings are studied. In particular, it is shown that forms a KA-radical class, where denotes the class of completely prime near-rings and the class of 3-prime near-rings. This gives another near-ring generalization of the 2-primal ring concept. The theory of radical pairs are also used to show that in general the class of 3-semiprime near-rings is not the semisimple class of the 3-prime radical.

A non-associative ring which contains a well-known associative ring or Lie ring is interesting. In this paper, a method to construct a Lie admissible non-associative ring is given; a class of simple non-associative algebras is obtained; all the derivations of the non-associative simple algebra defined in this paper are determined; and finally, a solid algebra is defined.

In this paper, we prove some basic properties of left weakly regular near-rings. We give an affirmative answer to the question whether a left weakly regular near-ring with left unity and satisfying the IFP is also right weakly regular. In the last section, we use among others left 0-prime and left completely prime ideals to characterize strongly regular near-rings.

In this paper, we give a concrete method to construct cellular algebras from matrix algebras by specifying certain fixed matrices for the data of inflations. In particular, orthogonal matrices can be chosen for such data.

In this paper, we introduce some Fuchsian groups and calculate their parabolic class numbers. In addition, we give two conjectures concerning with the parabolic class number.

Let R be a ring. For two fixed positive integers m and n, a right R-module M is called (m,n)-injective in case every right R-homomorphism from an n-generated submodule of R^m to M extends to one from R^m to M. R is said to be left (m,n)-coherent if each n-generated submodule of the left R-module R^m is finitely presented. In this paper, we give some new characterizations of (m,n)-injective modules. We also derive various equivalent conditions for a ring to be left (m,n)-coherent. Some known results on coherent rings are obtained as corollaries.

Let p be a prime number. In this paper, the author sets up a canonical correspondence between polynomial functions over ℤ/p²ℤ and 3-tuples of polynomial functions over ℤ/pℤ. Based on this correspondence, he proves and reproves some fundamental results on permutation polynomials mod p^l. The main new result is the characterization of strong orthogonal systems over ℤ/p^lℤ.

Finite groups in which each cyclic subnormal subgroup is Sylow-permutable, permutable or normal are investigated. Characterizations are found in both the soluble and insoluble cases.