We observe that the class of left and right artinian left and right morphic rings agrees with the class of artinian principal ideal rings. For R an artinian principal ideal ring and G a group, we characterize when RG is a principal ideal ring; for finite groups G, this characterizes when RG is a left and right morphic ring. This extends work of Passman, Sehgal and Fisher in the case when R is a...

In a recent paper it has been established that over an Artinian ring R all two-dimensional polynomial automorphisms having Jacobian determinant one are tame if R is a Q-algebra. This is a generalization of the famous Jung-Van der Kulk Theorem, which deals with the case that R is a field (of any characteristic). Here we will show that for tameness over an Artinian ring, the Q-algebra assumption ...

We investigate two invariants of Noetherian semiperfect rings, namely the depth and a new invariant we call “delooping level”. These give lower upper bounds for finitistic dimension, respectively. As first theorems, necessary sufficient criterion delooping level to be finite in terms splitting unit map related adjunction, use this torsionfreeness finiteness. further relate these Auslander–Bridg...

The Jacobson-Bourbaki Theorem for division rings was formulated in terms of corings by Sweedler in [14]. Finiteness conditions hypotheses are not required in this new approach. In this paper we extend Sweedler’s result to simple artinian rings using a particular class of corings, comatrix corings. A Jacobson-Bourbaki like correspondence for simple artinian rings is then obtained by duality.

We give a characterization of the Lefschetz elements in Artinian Gorenstein rings over a field of characteristic zero in terms of the higher Hessians. As an application, we give new examples of Artinian Gorenstein rings which do not have the strong Lefschetz property.

An easy consequence of this is that a left Noetherian (respectively left Artinian) ring which is finitely generated over its center is right Noetherian (respectively right Artinian). Theorem 1 follows easily from Theorem 2, which gives a partial converse to the following standard fact: IfR C S are rings, and ifQ is an injective R-module, then Hom~(S, Q) is an injective S-module (this follows, f...

Let Q = k[[x1, . . . , xn]] be the power series ring over a field k. Artinian Gorenstein quotients R = Q/I whose unique maximal ideal m satisfies ms 6= 0 = ms+1 are in correspondence via the Macaulay inverse system with degree s polynomials in n variables. Bøgvad constructed examples for which the Poincaré series of k over R is irrational. When s is even, we prove that such examples are rare. M...

For an arbitrary left Artinian ring R, explicit descriptions are given of all the left denominator sets S of R and left localizations SR of R. It is proved that, up to R-isomorphism, there are only finitely many left localizations and each of them is an idempotent localization, i.e. SR ≃ S e R and ass(S) = ass(Se) where Se = {1, e} is a left denominator set of R and e is an idempotent. Moreover...

Let R be a commutative local k-algebra of Krull dimension one, where k is field. α automorphism R, and define S to the skew polynomial algebra R[θ;α]. We offer, under some additional assumptions on criterion for have injective hulls all simple S-modules locally Artinian - that is, satisfy property (⋄). It easy well known if finite order, then has this property, but in order get when infinite we...

Given a direct sum G of cyclic groups, we find a sharp bound for the minimal number of proper subgroups whose union is G. This generalizes to sums of cyclic modules over more general rings, and we are able to solve this covering problem for all local, Artinian, or Dedekind rings (and even for monoids). We also minimally cover all finitely generated non-cyclic modules over quasi-local rings. The...