Unformatted text preview: b s . Show that p divides one of the b i . 6.5.21. Let aR be an ideal in a principal ideal domain R . Show that R=aR is a ring with only ﬁnitely many ideals. 6.5.22. (a) Let a be an element of an integral domain R . Show that R=aR is an integral domain if, and only if, a is prime. (b) Let a be an element of a principal ideal domain R . Show that R=aR is a a ﬁeld if, and only if, a is irreducible. (c) Show that a quotient R=I of a principal ideal domain R by a nonzero proper ideal I is an integral domain if, and only if, it is a ﬁeld. (d) Show that an ideal in a principal ideal domain is maximal if, and only if, it is prime. 6.5.23. Consider the ring of formal power series KŒŒxŁŁ with coefﬁcients in a ﬁeld K . (a) Show that the units of KŒŒxŁŁ are the power series with nonzero constant term (i.e., elements of the form ˛ C xf , where ˛ ¤ , and f 2 KŒŒxŁŁ )....
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 Fall '08
 EVERAGE
 Algebra, Integral domain, Ring theory, Unique factorization domain, Euclidean domain, Principal ideal domain

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