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Math Help - Principal ideal

  1. #1
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    Principal ideal

    Let Z be the integers and let I={5k| k=0,1,2,⋯} be the principal ideal generated by 5. Let a,b∈Z, show that a≡b (mod 5) if and only if a-b∈I.

    Need help starting this problem. do i show that I is a subset of Z?

    Because i know that a nonempty subset I of a ring R is an ideal iff it has these properties:
    i) if a,b are elements of I, then a-b is an element of I;
    ii) if r∈ R and a ∈ I, then ra ∈ I and ar ∈ I.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by empressA88 View Post
    Let Z be the integers and let I={5k| k=0,1,2,⋯} be the principal ideal generated by 5. Let a,b∈Z, show that a≡b (mod 5) if and only if a-b∈I.

    Need help starting this problem. do i show that I is a subset of Z?

    Because i know that a nonempty subset I of a ring R is an ideal iff it has these properties:
    i) if a,b are elements of I, then a-b is an element of I;
    ii) if r∈ R and a ∈ I, then ra ∈ I and ar ∈ I.
     a\equiv b\bmod{5} \iff 5\mid a-b \iff a-b=5k\iff a-b\in I
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