It is unknown if an Artinian level O-sequence of codimension 3 and type r (≥ 2) is unimodal, while it is known that any Gorenstein O-sequence of codimension 3 is unimodal. We show that some Artinian non-unimodal O-sequence of codimension 3 cannot be level. We also find another non-level case: if some Artinian algebra A of codimension 3 has the Hilbert function H : h0 h1 · · · hd−1 hd · · · hd }...

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...

Let [Formula: see text] be a commutative Noetherian ring. For finitely generated text]-module text], Northcott introduced the reducibility index of which is number submodules appearing in an irredundant irreducible decomposition submodule text]. On other hand, for Artinian Macdonald proved that sum-irreducible representation does not depend on choice representation. This called sum-reducibility...

Let R be a commutative ring. An R-module M is called co-multiplication provided that foreach submodule N of M there exists an ideal I of R such that N = (0 : I). In this paper weshow that co-multiplication modules are a generalization of strongly duo modules. Uniserialmodules of finite length and hence valuation Artinian rings are some distinguished classes ofco-multiplication rings. In additio...

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 ...

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.

Let ?: R0?R be a ring homomorphism and suppose that a and a0, respectively, are ideals of R and R0 such that is an Artinian ring. Let M and N be two finitely generated R-modules and suppose that (R0,m0) is a local ring. In this note we prove that the R-modules and are Artinian for all integers i and j, whenever and . Also we will show that if a is principal, then the R-modules and ...