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doi: 10.1080/00927879208824455pmid: N/A
A unital left R-module R M is said to have property (S) if every surjective endomorphism of R M is an automorphism, the ring R is called left (right) S-ring if every left (right) R-module with property (S) is Noetherian, R is called S-ring if it is both a left and a right S-ring. In this note we show that a duo ring is a left S-ring if and only if it is left Artinian left principal ideal ring. To do this we shall construct on every non distributive Artinian local ring with radical square zero a non-finitely generated module with property (S). And we give an example of left duo left Artinian left principal ideal ring which is not a left S-ring, showing the necessity of the ring to be duo in the above result.
doi: 10.1080/00927879208824456pmid: N/A
Let k be a field of characteristic p>0 and D≠0 a family of k-derivations of k[x,y]. It is proved in [1] that k[x,y]D, the ring of constants with respect to D, can be generated, as a k[x p,y p]-algebra, by p - 1 elements. In this note we prove that p - 1 is the sharp upper bound of numbers of generators.
doi: 10.1080/00927879208824460pmid: N/A
The notion of a simple ring DGderived from a group ring KG is introduced in case K is a field and G is an infinite residually finite group. The close link between DGand KG is exploited in both directions: first, for a simple proof of the Kaplansky's conjecture concerning direct finiteness of KG. Second, to show that DGprovides counter-examples to some conjectures dealing with von Neumann regular rings and the rings all of whose one-sided ideals are generated by idempotents.
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