This is my first question; I hope it worthy of this awesome forum and its members.
Let R be a commutative ring, perhaps with unit, perhaps not. As usual let R[x]
denote the ring of polynomials over R, and let I be an ideal in R[x]. Let
D(I) be the subset of I consisting of all f(x)inI of minimal degree. It is
well known from elementary algebra that in the event that R is a field, D(I)
consists of essentially one element d(x) (up to multiplication by a unit of R),
and that I is the principal ideal generated by d(x): I=(d(x))=Rd(x).
In the case of general commutative R, does D(I) generate I in the sense that
any f(x)inI may be written f(x)=summi=1fi(x)di(x) with di(x)inD(I) and fi(x)inR[x]? Does the existence of 1inR make any difference?
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