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Math Help - Integral Domain

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    MHF Contributor chiph588@'s Avatar
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    Integral Domain

    If  R is an integral domain and  a, b \in R , show if  Ra = Rb then  a=ub where  u is a unit.
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    Quote Originally Posted by chiph588@ View Post
    If  R is an integral domain and  a, b \in R , show if  Ra = Rb then  a=ub where  u is a unit.
    Since (a) = (b) it means (a)\subseteq (b) \text{ and }(b) \subseteq (a).
    Thus, b|a \text{ and } a|b so a,b are associates and so a=ub.
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    Quote Originally Posted by chiph588@ View Post

    If  R is an integral domain and  a, b \in R , show if  Ra = Rb then  a=ub where  u is a unit.
    to see why we need R to be an integral domain:

    if a=0, then Rb=0 and so b=0. in this case choose u=1. so we may assume that a \neq 0. now we have a \in Ra = Rb. so a=rb, for some r \in R. similarly b=sa, for some s \in R.

    thus a=rb=rsa. hence (1-rs)a=0. since R is an integral domain and a \neq 0, we must have 1-rs=0, i.e. rs=1. so r is a unit and we choose u=r.
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