Transcription of Solutions to Homework Problems from Chapter 3
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Solutions to Homework Problems from Chapter 3
Solutions to Homework Problems from Chapter 3 The following subsets ofZ(with ordinary addition and multiplication) satisfy all but one of theaxioms for a ring. In each case, which axiom fails.(a) The setSof odd integers. The sum of two odd integers is a even integer. Therefore, the setSis not closed under , Axiom 1 is violated.(b) The set of nonnegative integers. Ifais a positive integer, then there is no solution ofa+x= 0 that is also positive. Hence, Axiom5 is (a) Show that the setRof all multiples of 3 is a subring ofZ.(b) Letkbe a fixed integer. Show that the set of all multiples ofkis a subring ofZ. Clearly, (b) implies (a); so let us just prove (b). LetS={z Z|z=nkfor somen Z}.In general, to show that a subsetSof a ringR, is a subring ofR, it is sufficient to show that(i)Sis closed under addition inR(ii)Sis closed under multiplication inR;(iii) 0R S;(iv) whena S, the equationa+x= 0 Rhas a solution , b, c S Zwitha=rk,b=sk,c=tk.
Solutions to Homework Problems from Chapter 3 §3.1 3.1.1. The following subsets of Z (with ordinary addition and multiplication) satisfy all but one of the axioms for a ring. In each case, which axiom fails. (a) The set S of odd integers. • The sum of two odd integers is a even integer. Therefore, the set S is not closed under addition.
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