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Abstract: In this paper, we investigate some basic classical results on the Krull dimension of -graded commutative rings by using the concept of iterated boundary ideals and iterated boundary monoids. After having pointed out the link between the Krull dimension (Kdim) and the boundary ideals and monoids, we characterize in the same way the graded dimension (dimgr). The main goal is to compare , and , where is a graded ring (noetherian or not). To do this, we establish various properties using the boundary ideals, in particular some typical boundary ideals inclusions. [ABSTRACT FROM AUTHOR]

Abstract: This paper shows a mean (using multilinear alternating forms) of getting unimodular vectors in a module over a commutative ring, without noetheriannity hypothesis. We show an elementary approach of Serre''s splitting off theorem, Bass''s stable range and cancellation theorems and Forster–Swan''s theorem. [Copyright &y& Elsevier]