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Rings with a weak algorithm

1963; American Mathematical Society; Volume: 109; Issue: 2 Linguagem: Inglês

10.1090/s0002-9947-1963-0153696-8

1088-6850

P. M. Cohn,

Polynomial and algebraic computation

Commutative principal ideal domains form a somewhat special class of rings, which, however, possess many pleasant properties.Often we can make the task of deriving these properties even easier by limiting ourselves to Euclidean domains.If we have a valuated (commutative) ring and ask that it shall be a Euclidean domain with respect to this valuation, the ring is almost uniquely determined: it must be a polynomial ring F[x] in a single indeterminate over a field F.All these ideas can be generalized in a straightforward manner to the noncommutative case : the principal ideal domains again form a well-behaved though rather narrow class (cf.Jacobson [9, Chapter 3]), and the valuated rings with a Euclidean algorithm are just the skew polynomial rings fc[x;S,D] over a skew field fc with an automorphism S and an S-derivation D (2).One obtains a slightly larger class by taking, instead of principal ideal domains, Bezout rings, i.e., integral domains in which any finitely generated (left or right) ideal is principal but this probably amounts to not much more than allowing locally principal ideal domains.A significantly wider class of rings is obtained by taking all integral domains in which any two principal right ideals with a nonzero intersection have a sum and intersection which are again principal.These are the weak Bezout rings introduced in [6], where it is shown that a weak Bezout ring in which prime factorizations exist, is a unique factorization domain, and other decomposition theorems hold (corresponding to the primary decomposition of an ideal in a Noetherian ring).Further it is shown there that the weak Bezout rings include free associative algebras in any number of free generators over a field.It is possible to weaken the definition of the Euclidean algorithm in a similar way so as to obtain rings with a weak algorithm (cf.§2 for the definition).This was first introduced in [4] where it was applied to prove (in effect) that in any ring R with a weak algorithm, all right ideals were free R-modules.We now continue the study of rings with a weak algorithm and in particular show that they are weak Bezout rings, so that the results of [6] become applicable ( §4).This