We study symmetric algebras $A$ over a field $F$ in which the Jacobson radical of center $A$, socle or Reynolds ideal are ideals.

We study representations of nilpotent type nontrivial liftings of quantum linear spaces and their Drinfel’d quantum doubles. We construct a family of Vermatype modules in both cases and prove a parametrization theorem for the simple modules. We compute the Loewy and socle series of Verma modules under a mild restriction on the datum of a lifting. We find bases and dimensions of simple modules.

A frame $L$ is called {it coz-dense} if $Sigma_{coz(alpha)}=emptyset$ implies $alpha=mathbf 0$. Let $mathcal RL$ be the ring of real-valued continuous functions on a coz-dense and completely regular frame $L$. We present a description of the socle of the ring $mathcal RL$ based on minimal ideals of $mathcal RL$ and zero sets in pointfree topology. We show that socle of $mathcal RL$ is an essent...

We study representations of nilpotent type nontrivial liftings of quantum linear spaces and their Drinfel’d quantum doubles. We construct a family of Vermatype modules in both cases and prove a parametrization theorem for simple modules. We compute the Loewy and socle series of Verma modules under a mild restriction on the datum of a lifting. We find bases and dimensions of simple modules.

A complete extension theorem for linear codes over a module alphabet and the symmetrized weight composition is proved. It is shown that an extension property with respect to arbitrary weight function does not hold for module alphabets with a noncyclic socle.