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It is well-known that a ring R is right Noetherian if and only if every direct sum of injective right R-modules is injective. In this paper, we characterize Nonnil-Noetherian rings by nonniMnjective modules. If R is a commutative ring, then we prove that R is a Nonnil-Noetherian ring if and only if the ring R[x]/(xn+1) is Nonnil-Noetherian, where (xn+1) is the ideal generated by xn+1 and n is a non-negative integer. [ABSTRACT FROM AUTHOR]

Let R be a ring, G an ordered group and a a map from G into the group of automorphisms of R. If R is reduced, then the Malcev-Nemnann ring R * ((G)) has the same triangulating dimension as R. Furthermore, if R is a PWP ring, then so is R * ((G)). [ABSTRACT FROM AUTHOR]