More on the Schur group of a commutative ring.
Let be a Banach space of dimension and be a standard operator algebra. In the present paper it is shown that if a mapping (not necessarily linear) satisfies for all , then , where is an additive derivation of and vanishes at second commutator for all . Moreover, if is linear and satisfies the above relation, then there exists an operator and a linear mapping from into satisfying for all , such that for all .
Let be a graded ring and be an integer. We introduce and study the notions of Gorenstein -FP-gr-injective and Gorenstein -gr-flat modules by using the notion of special finitely presented graded modules. On -gr-coherent rings, we investigate the relationships between Gorenstein -FP-gr-injective and Gorenstein -gr-flat modules. Among other results, we prove that any graded module in -gr (or gr-) admits a Gorenstein -FP-gr-injective (or Gorenstein -gr-flat) cover and preenvelope, respectively....
Let be a graded ring and an integer. We introduce and study -strongly Gorenstein gr-projective, gr-injective and gr-flat modules. Some examples are given to show that -strongly Gorenstein gr-injective (gr-projective, gr-flat, respectively) modules need not be -strongly Gorenstein gr-injective (gr-projective, gr-flat, respectively) modules whenever . Many properties of the -strongly Gorenstein gr-injective and gr-flat modules are discussed, some known results are generalized. Then we investigate...
The Hopf algebra of word-quasi-symmetric functions (), a noncommutative generalization of the Hopf algebra of quasi-symmetric functions, can be endowed with an internal product that has several compatibility properties with the other operations on . This extends constructions familiar and central in the theory of free Lie algebras, noncommutative symmetric functions and their various applications fields, and allows to interpret as a convolution algebra of linear endomorphisms of quasi-shuffle...
We give conditions for the skew group ring S * G to be strongly separable and H-separable over the ring S. In particular we show that the H-separability is equivalent to S being central Galois extension. We also look into the H-separability of the ring S over the fixed subring R under a faithful action of a group G. We show that such a chain: S * G H-separable over S and S H-separable over R cannot occur, and that the centralizer of R in S is an Azumaya algebra in the presence of a central element...
Let be an infinite-dimensional complex Hilbert space and be a standard operator algebra on which is closed under the adjoint operation. It is shown that every nonlinear -Lie higher derivation of is automatically an additive higher derivation on . Moreover, is an inner -higher derivation.
Weakly associative lattice rings (wal-rings) are non-transitive generalizations of lattice ordered rings (l-rings). As is known, the class of l-rings which are subdirect products of linearly ordered rings (i.e. the class of f-rings) plays an important role in the theory of l-rings. In the paper, the classes of wal-rings representable as subdirect products of to-rings and ao-rings (both being non-transitive generalizations of the class of f-rings) are characterized and the class of wal-rings having...
We give some criteria of normability of an S-ring, and we study the properties of its norms.