Let SI be a finite dimensional algebra with unit element over a field. 21 is generalized uniserial if every primitive (left or right) ideal has a unique composition series. 21 is a UMFR algebra (an algebra with a unique minimal faithful representation) if 21 has only one faithful representation which is minimal with respect to being faithful. The notation used in this paper will be that of an e...

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It is proved that a ring A right or left Noetherian, distributive, centrally essential if and only [Formula: see text], where each of the rings text] either commutative Dedekind domain Artinian, uniserial ring.

let $r=oplus_{nin bbb n_0}r_n$ be a noetherian homogeneous ring with local base ring $(r_0,frak{m}_0)$, $m$ and $n$ two finitely generated graded $r$-modules. let $t$ be the least integer such that $h^t_{r_+}(m,n)$ is not minimax. we prove that $h^j_{frak{m}_0r}(h^t_{r_+}(m,n))$ is artinian for $j=0,1$. also, we show that if ${rm cd}(r_{+},m,n)=2$ and $tin bbb n_0$, then $h^t_{frak{m}_0r}(h^2_{...

We describe of all finite dimensional uniserial representations of a commutative associative (resp. abelian Lie) algebra over a perfect (resp. sufficiently large perfect) field. In the Lie case the size of the field depends on the answer to following question, considered and solved in this paper. Let K/F be a finite separable field extension and let x, y ∈ K. When is F [x, y] = F [αx+ βy] for s...