Tuesday, April 30, 2013

Principle Ideals in Integral Domains (7.4.8)

Dummit and Foote Abstract Algebra, section 7.4, exercise 8:

MathJax TeX Test Page Let R be an integral domain with a,bR. Prove (a)=(b) if and only if a=ub for some unit u.

Proof: () We clearly have a(b)=Rb, and since u1a=u1(ub)=b, we have b(a)=Ra. Therefore, Ra=(a)(b) and Rb=(b)(a) so that (a)=(b). () If either of a or b are zero, then both are zero, and the case clearly holds, so assume a0b. We must have a=ub and va=b for some u,vR. We can compare the two to obtain va=vub=b, so that vubb=(vu1)b=0, and now vu1=0 implying vu=uv=1, i.e. u is a unit as claimed. 

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