by

DOI: 10.1023/A:1017213903245
Print publication date: 6/1/1999
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by Chen, Huanyin

We prove that every exchange ring with primitive factors Artinian is clean. Also, it is shown that for exchange rings with Artinian primitive factors, the following are equivalent: (1) Every element in R is a sum of two units. (2) There exist α, β ∈ U(R) such that α + β = 1. (3) R does not have Z / 2 Z as a homomorphic image. Finally, we prove that exchange ring R is strongly π-regular if the Jacobson radical of any homomorphic image of R is T-nilpotent or locally nilpotent. These are generalizations of the corresponding results of A. Badawi, W. D. Burgess and P. Menal, Fisher and Snider, and J. Stock.

DOI: 10.1023/A:1009927211591
Print publication date: 6/1/1999
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by Ene, Viviana; Popescu, Dorin

Let k be an algebraically closed field of characteristic ≠ 3 and i, j, t some positive integers such that 1 ≤ i < j < t, i + j ≠ t. Then there exist a finite number of nonisomorphic indecomposable maximal Cohen–Macaulay modules N over k[[x, y]] /(x^t + y³) such that N / y N is a direct sum of copies of k[[x]] /(xⁱ), k[[x]] /(x^j) and we describe them completely.

DOI: 10.1023/A:1009942810682
Print publication date: 6/1/1999
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by Militaru, G.

Let H be a bialgebra and _H L^H be the category of Long H-dimodules defined, for a commutative and co-commutative H, by F. W. Long and studied in connection with the Brauer group of a so-called H-dimodule algebra. For a commutative and co-commutative H, _H L^H =_H YD^H (the category of Yetter–Drinfel’d modules), but for an arbitrary H, the categories _H L^H and _H YD^H are basically different. Keeping in mind that the category _H YD^H is deeply involved in solving the quantum Yang–Baxter equation, we study the category _H L^H of H-dimodules in connection with what we have called the D-equation: R¹² R²³ = R²³ R¹², where R ∈ End_k(M ⊗ M) for a vector space M over a field k. The main result is a FRT-type theorem: if M is finite-dimensional, then any solution R of the D-equation has the form R = R_{(M, ⋅, ρ)}, where (M, ⋅, ρ) is a Long D(R)-dimodule over a bialgebra D(R) and R_{(M, ⋅, ρ)} is the special map R_{(M, ⋅, ρ)}(m ⊗ n) : = ∑ n_〈1〉 ⋅ m ⊗ n_〈0〉. In the last section, if C is a coalgebra and I is a coideal of C, we introduce the notion of D-map on C, that is a k-bilinear map σ : C ⊗ C / I → k satisfying a condition which ensures on the one hand that, for any right C-comodule, the special map R_σ is a solution of the D-equation and, on the other, that, in the finite case, any solution of the D-equation has this form.

DOI: 10.1023/A:1009905324871
Print publication date: 6/1/1999
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by Linckelmann, Markus

We develop the notion of a cohomology ring of blocks of finite groups and study its basic properties by means of transfer maps between the Hochschild cohomology rings of symmetric algebras associated with bounded complexes of finitely generated bimodules which are projective on either side.

DOI: 10.1023/A:1009979222100
Print publication date: 6/1/1999
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