The Anderson-Badawi conjecture for commutative algebras over infinite fields

In [1], Anderson and Badawi conjecture that every n-absorbing ideal of a commutative ring is strongly n-absorbing. In this article we prove their conjecture in certain cases (in particular this is the case for commutative algebras over an infinite field). We also show that an affirmative answer to another conjecture in [1] implies the Anderson-Badawi Conjecture.     

Monomial Hopf algebras over fields of positive characteristic

In this paper, we study the structures of monomial Hopf algebras over a field of positive characteristic. A necessary and sufficient condition for the monomial coalgebra Cd(n) to admit Hopf structures is given here, and if it is the case, all graded Hopf structures on Cd(n) are completely classified. Moreover, we construct a Hopf algebras filtration on Cd(n) which will help us to discuss a conjecture posed by Andruskiewitsch and Schneider. Finally combined with a theorem by Montgomery, we give the structure theorem for all monomial Hopf algebras.     

Quaternion algebras over commutative rings

Translated from Matematicheskie Zametki, Vol. 53, No. 2, pp. 126–131, February, 1993.     

On commutative group algebras of mixed groups

Suppose F is a perfect field of characteristic p 0 and G is an abelian group whose torsion subgroup Gt is p-primary. If Gt is totally projective of countable length, it is shown that G is a direct factor of the group of normalized units V(G) of the group algebra F(G) and that V(G)/G is a totally projective p-group. The proof of this result is based on a new characterization of the class of totally projective p-groups of countable length. Li addition, the same result regarding V(G) is obtained if G has countable torsion-free rank and Gt is totally projective of length less than ω₁ + ω₀ . Finally, these results are applied to the question of whether the existence of an F-i pomorphism F(G) ? F(H) for some group H implies that G?H.     

On the isomorphism of commutative group algebras

LetG andH be finite abelian groups and letF be an arbitrary field. One fundamental problem is that of determining necessary and sufficient conditions for the isomorphism of the group algebrasFG andFH. No solution has appeared in the literature. Nevertheless by combining the results of Berman, Perlis and Walker, Cohen, and Deskins and providing connecting arguments a complete solution can be obtained. It is the purpose of this note to present such a solution.     

Automorphisms of matrix algebras over commutative rings

The automorphisms of an n × n matrix algebra over a commutative ring can fail to be inner. The extent of this failure, however, is under control. For instance, the commutator of any two automorphisms and the nth power of each of them are necessarily inner.     

Spectral synthesis in Banach modules over commutative banach algebras

Mathematical Notes -     

Grassmann algebras over finite fields

We study an analogue of a problem of procesi about matrices [9, page 185(e)]: are there non-trivial polynomials over Z which become identities over Z _p - for grassmann algebras E? when 1 ? E, we show that such polynomials do not exist, but when 1 ?,E such polynomials exist - also for matrices over E. these results are deduced from a careful study of the various codimensions of these algebras.     

Reduction of derived Hochschild functors over commutative algebras and schemes

We study functors underlying derived Hochschild cohomology, also called Shukla cohomology, of a commutative algebra S essentially of finite type and of finite flat dimension over a commutative noetherian ring K. We construct a complex of S-modules D, and natural reduction isomorphisms for all complexes of S-modules N and all complexes M of finite flat dimension over K whose homology H(M) is finitely generated over S; such isomorphisms determine D up to derived isomorphism. Using Grothendieck duality theory we establish analogous isomorphisms for any essentially finite-type flat map of noetherian schemes, with f^!OY in place of D.