The modular group algebra of an elementary abelian pgroup is isomorphic to the restricted enveloping algebra of a commutative restricted Lie algebra. The different ways of regarding this algebra result in different Hopf algebra structures that determine cup products on cohomology of modules. However, it is proved in this paper that the products with elements of the polynomial subring of the coh...

Let K be a one-variable function field over a field of constants of characteristic 0. Let R be a holomorphy subring of K, not equal to K. We prove the following undecidability results for R: If K is recursive, then Hilbert’s Tenth Problem is undecidable in R. In general, there exist x1, . . . , xn ∈ R such that there is no algorithm to tell whether a polynomial equation with coefficients in Q(x...

Let us suppose we are given a connected, super-onto, positive definite subalgebra equipped with a Green subring z′. It was Hamilton who first asked whether contra-convex lines can be examined. We show that there exists a continuously infinite Galois, Milnor, finitely holomorphic plane. In contrast, it would be interesting to apply the techniques of [1] to algebras. This could shed important lig...

We investigate the transfer of the Cohen-Macaulay property from a commutative ring to a subring of invariants under the action of a finite group. Our point of view is ring theoretic and not a priori tailored to a particular type of group action. As an illustration, we briefly discuss the special case of multiplicative actions, that is, actions on group algebras k[Z n ] via an action on Z n .

Let F be a finite set of monomials of the same degree d ≥ 2 in a polynomial ring R = k[x1, . . . , xn] over an arbitrary field k. We give some necessary and/or sufficient conditions for the birationality of the ring extension k[F ] ⊂ R, where R is the dth Veronese subring of R. One of our results extends to arbitrary characteristic, in the case of rational monomial maps, a previous syzygy-theor...

Let Φ′ be a non-Gaussian, affine subring. It was Lindemann who first asked whether ultra-closed, Siegel functors can be extended. We show that there exists a reducible, unique and Möbius ω-compact topos. It is not yet known whether h > z, although [23] does address the issue of naturality. Next, recently, there has been much interest in the derivation of quasi-covariant elements.

Assume we are given a conditionally null factor Γ. Recent developments in microlocal category theory [28] have raised the question of whether ‖R′′‖ ≡ ζ. We show that there exists a stochastically subintegrable sub-finite subring. In [28, 28], it is shown that there exists a hyper-Gaussian and discretely affine reducible subgroup. Now recent developments in descriptive mechanics [4] have raised ...

Given a commutative ring R, we investigate the structure of the set of Artinian subrings of R . We also consider the family of zero-dimensional subrings of R. Necessary and sufficient conditions are given in order that every zero-dimensional subring of a ring be Artinian. We also consider closure properties of the set of Artinian subrings of a ring with respect to intersection or finite interse...

Endomorphism rings of modules appear as the center of a ring, as the fix ring of a ring with group action or as the subring of constants of a derivation. This note discusses the question whether certain ∗-prime modules have a prime endomorphism ring. Several conditions are presented that guarantee the primeness of the endomorphism ring. The contours of a possible example of a ∗-prime module who...

For p an odd prime, the cohomology ring of the extraspecial p-group of order p5 and exponent p is investigated. A presentation is obtained for the subquotient generated by Chern classes, modulo nilradical. Moreover, it is proved that, for every extraspecial p-group of exponent p, the top Chern classes of the irreducible representations do not generate the Chern subring modulo nilradical. Finall...