All commutative semigroups $S$ are described such that the Jacobson radical is homogeneous in each ring graded by $S$.

For any non-torsion group $G$ with identity $e$, we construct a strongly $G$-graded ring $R$ such that the Jacobson radical $J\left(R_e\right)$ is locally nilpotent, but $J\left(R\right)$ is not locally nilpotent. This answers a question posed by Puczyłowski.

Let $G$ be a semisimple complex algebraic group and $X$ its flag variety. Let $𝔤=\mathrm{Lie}\left(G\right)$ and let $U$ be its enveloping algebra. Let $𝔥$ be a Cartan subalgebra of $𝔤$. For $\mu \in {𝔥}^{*}$, let $J_{\mu }$ be the corresponding minimal primitive ideal, let $U_{\mu }=U/J_{\mu }$, and let ${𝒯}_{U_{\mu }}:K_0(U_{m}u)\to ℂ$ be the Hattori-Stallings trace. Results of Hodges suggest to study this map as a step towards a classification, up to isomorphism or Morita equivalence, of the $ℂ$-algebras $U_{\mu }$. When $\mu$ is regular, Hodges has shown that $K_0(U_{\mu })\cong K_0\left(X\right)$. In this case $K_0(U_{\mu })$ is generated by the classes corresponding to...

Let Λ be a tubular algebra over an arbitrary base field. We study the Grothendieck group $K_0\left(\Lambda \right)$, endowed with the Euler form, and its automorphism group $Aut\left(K_0\left(\Lambda \right)\right)$ on a purely K-theoretical level as in [7]. Our results serve as tools for classifying the separating tubular families of tubular algebras as in the example [5] and for determining the automorphism group $Aut\left(D^b\Lambda \right)$ of the derived category of Λ.

Let S be a semiring whose additive reduct (S,+) is an inverse semigroup. The relations θ and k, induced by tr and ker (resp.), are congruences on the lattice C(S) of all congruences on S. For ρ ∈ C(S), we have introduced four congruences ${\rho }_{min},{\rho }_{max},{\rho }^{min}$ and ${\rho }^{max}$ on S and showed that $\rho \theta =[{\rho }_{min},{\rho }_{max}]$ and $\rho \kappa =[{\rho }^{min},{\rho }^{max}]$. Different properties of ρθ and ρκ have been considered here. A congruence ρ on S is a Clifford congruence if and only if ${\rho }_{max}$ is a distributive lattice congruence and ${\rho }^{max}$ is a skew-ring congruence on S. If η (σ) is the least distributive...

Given a number field $K$ Galois over the rational field $\mathbb{Q}$, and a positive integer $n$ prime to the class number of $K$, there exists an abelian extension $L/K$ (of exponent $n$) such that the $n$-torsion subgroup of the Brauer group of $K$ is equal to the relative Brauer group of $L/K$.