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We will consider unital rings A with free additive group, and want to construct (in ZFC) for each natural number k a family of -free A-modules G which are separable as abelian groups with special decompositions. Recall that an A-module G is -free if every subset of size is contained in a free submodule (we will refine this in Definition 3.2); and it is separable as an abelian group if any finite subset of G is contained in a free direct summand of G. Despite the fact that such a module G is...
Let X and Y be complex Banach spaces of dimension greater than 2. We show that every 2-local Lie isomorphism ϕ of B(X) onto B(Y) has the form ϕ = φ + τ, where φ is an isomorphism or the negative of an anti-isomorphism of B(X) onto B(Y), and τ is a homogeneous map from B(X) into ℂI vanishing on all finite sums of commutators.
2000 Mathematics Subject Classification: 16R10, 16R20, 16R50The algebra Mn(K) of the matrices n × n over a field K can be regarded as a Z-graded algebra. In this paper, it is proved that if K is an
infinite field, all the Z-graded polynomial identities of Mn(K) follow from the identities:
x = 0, |α(x)| ≥ n,
xy = yx, α(x) = α(y) = 0,
xyz = zyx, α(x) = −α(y) = α(z ),
where α is the degree of the corresponding variable.
This is a generalization of a result of Vasilovsky about the Z-graded identities...
Let be the set of nonnegative integers and the ring of integers. Let be the ring of matrices over generated by the following two matrices: one obtained from the identity matrix by shifting the ones one position to the right and the other one position down. This ring plays an important role in the study of directly finite rings. Calculation of invertible and idempotent elements of yields that the subrings generated by them coincide. This subring is the sum of the ideal consisting of...
Based on the work of D. Happel, I. Reiten and S. Smalø on quasitilted artin algebras, the first two authors recently introduced the notion of quasitilted rings. Various authors have presented examples of quasitilted artin algebras that are not tilted. Here we present a class of right quasitilted rings that not right tilted, and we show that they satisfy a condition that would force a quasitilted artin algebra to be tilted.
In continuation of our recent work about smash product Hom-Hopf algebras [Colloq. Math. 134 (2014)], we introduce the Hom-Yetter-Drinfeld category via the Radford biproduct Hom-Hopf algebra, and prove that Hom-Yetter-Drinfeld modules can provide solutions of the Hom-Yang-Baxter equation and is a pre-braided tensor category, where (H,β,S) is a Hom-Hopf algebra. Furthermore, we show that is a Radford biproduct Hom-Hopf algebra if and only if (A,α) is a Hom-Hopf algebra in the category . Finally,...
Suppose is a field of characteristic and is a -primary abelian -group. It is shown that is a direct factor of the group of units of the group algebra .
We find examples of polynomials whose eigenring is a central simple algebra over the field .
We first propose a generalization of the notion of Mathieu subspaces of associative algebras
, which was introduced recently in [Zhao W., Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra, 2010, 214(7), 1200–1216] and [Zhao W., Mathieu subspaces of associative algebras], to
-modules
. The newly introduced notion in a certain sense also generalizes the notion of submodules. Related with this new notion, we also introduce the sets σ(N) and τ(N) of stable...
If and are two Brzeziński crossed products and Q: W⊗ V → V⊗ W is a linear map satisfying certain properties, we construct a Brzeziński crossed product . This construction contains as a particular case the iterated twisted tensor product of algebras.
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