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A remark on power series rings

Publicacions Matemàtiques
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  • Computer Science


Publicacions Matemátiques, Vol 36 (1992), 481--484. Abstract A REMARK ON POWER SERIES RINGS P . M. COHN In memory of Pere Menal A trivializability principle for local rings is described which leads to a form of weak algorithm for local semifirs with a finitely gen- erated maximal ideal whose powers meet in zero . 1 . The ring of power series in several non-commuting indeterminates over a field has been characterized by the inverse weak algorithm relative to a suitable filtration (cf. [1], [4]) . As an example one may take a fir R with an ideal a such that R/a is a skew field ([4, Cor . 2 .9.16]) ; the filtration in this case is the a-adic filtration . This shows that any fir R which is a local ring with maximal ideal m satisfies the inverse weak algorithm relative to the m-adic filtration . Of course the power series ring itself is not a fir, but we get a fir by taking the subring of all rational power series Q4, p . 460] ) . However, the proof in [4] does not extend to semifirs, or even to one- sided firs . Our object here is to describe a similar result for semifirs ; the only additional hypothesis needed is that the ring is local with a maximal ideal which is finitely generated, as right ideal and whose powers intersect in zero . We shall also describe a form of trivializability for local rings (in Th. 1) which does not seem to have been noticed before . 2 . Throughout, all rings are associative, with unit element 1 :~¿ 0 (Le . non-trivial) ; undefined terms are as in [4] . We recall from [2, Th. 1] that any left or right semihereditary local ring is a semifir . Such rings satisfy the following strong dependence condition : D. Given al, . . . , an E R, if the al are right linearly dependent over R: aibi = 0, bi E R, bi not all 0, 482 P. M. COHN then the al satisfy a relation (1) in which one of the coefficients bi is unit . It is clear that any non-trivial ring satisfying D is a semifir . More gen- erally, we have the following result, which describe

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