Abstract. An integral domain is IDF if every non-zero element has only finitely many non-associate irreducible divisors. We investigate when R IDF implies that the ring of polynomials R[T ] is IDF. This is true when R is Noetherian and integrally closed, in particular when R is the coordinate ring of a non-singular variety. Some coordinate rings R of singular varieties also give R[T ] IDF. Analogous results for the related concept of IDPF are also given. The main result on IDF in this paper states that every countable domain embeds in another countable domain R such that R has no irreducible elements, hence vacuously IDF, and the polynomial ring R[T ] is not IDF. This resolves an open question. It is also shown that some subrings R of the ring of Gaussian integers known to be IDPF also have the property that R[T ] is not IDPF.
Abstract. A pair of complex vector spaces (V, W) is called a system if and only if there is a C-bilinear map from C2 X V to W. The category of systems contains subcategories equivalent to the category of modules over the ring of complex polynomials. Many concepts in the latter generalize to the category of systems. In this paper the pure projective systems are characterized and a bound on the rank of purely simple systems is obtained.
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