Let be a coalgebra over a field k. We introduce an operator Tr that takes a right quasi-finitely copresented -comodule M to a left quasi-finitely copresentedcomodule Tr M. If M is indecomposable not injective and Tr M is finite-dimensional over k, we prove the existence of an almost split sequence 0 → M → E → DTr M → 0 in the category of all right -comodules, where D = Hom k k . If is right semiperfect and the embedding of each simple right comodule S into its injective envelope I S has the property that the socle of I S /S is finite-dimensional, the above almost split sequence exists for each finite-dimensional M, and DTr M is also finite-dimensional.  2002 Elsevier Science (USA)

Abstract. A classical theorem of Burnside asserts that if % is a faithful complex character for the finite group G , then every irreducible character of G is a constituent of some power x" of X ■ Fifty years after this appeared, Steinberg generalized it to a result on semigroup algebras K[G] with K an arbitrary field and with G a semigroup, finite or infinite. Five years later, Rieffel showed that the theorem really concerns bialgebras and Hopf algebras. In this note, we simplify and amplify the latter work.Let K be a field and let A be a .£-algebra. A map A: A -► A ® A is said to be a comultiplication on A if A is a coassociative A^-algebra homomorphism. For convenience, we call such a pair (A, A) a b-algebra. Admittedly, this is rather nonstandard notation. One is usually concerned with bialgebras, that is, algebras which are endowed with both a comultiplication A and a counit e:A -* K. However, semigroup algebras are not bialgebras in general, and the counit rarely comes into play here. Thus it is useful to have a name for this simpler object. Proof. Certainly / is an ideal of A . Now let X = ®J2V€Sr V he the direct sum of the modules in y. Then X is an ,4-module and ann^ X = f)Ve^-aaaA V =

ABSTRACT. We use the duality between group gradings and group actions to study polycyclic-by-finite group-graded rings. We show that, for such rings, graded Noetherian implies Noetherian and relate the graded Krull dimension to the Krull dimension. In addition we find a bound on the length of chains of prime ideals not containing homogeneous elements when the grading group is nilpotent-by-finite.These results have suitable corollaries for strongly groupgraded rings. Our work extends several results on skew group rings, crossed products and group-graded rings.

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We give an alternative construction of the duality between finite group actions and group gradings on rings which was shown by Cohen and Montgomery in [1]. This duality is then used to extend known results on skew group rings to corresponding results for large classes of group-graded rings. Finally we modify the construction slightly to handle infinite groups.