# Answer to Question #12544 in Abstract Algebra for Hym@n B@ss

Question #12544

Prove that in principal ideal ring for every pair of elements exists their GCD.

Prove that if d=GCD(a,b), then there are such elements u,v that d=au+bv.

Prove that if d=GCD(a,b), then there are such elements u,v that d=au+bv.

Expert's answer

Let d=GCD(a,b)

Then aR < dR, bR < dR. So, aR+bR < dR.

As R is

principal ideal ring, then aR+bR=cR, for some c in R.

Then a*1+b*0=cu, for

some u, hence c|a. Analogously, c|b. So c|d by definition.

Then, aR+bR = cR

> dR > aR+bR . Result: aR+bR=dR, where d=GCD(a,b).

Then there are such

elements u,v in R, that au+bv=d.

Then aR < dR, bR < dR. So, aR+bR < dR.

As R is

principal ideal ring, then aR+bR=cR, for some c in R.

Then a*1+b*0=cu, for

some u, hence c|a. Analogously, c|b. So c|d by definition.

Then, aR+bR = cR

> dR > aR+bR . Result: aR+bR=dR, where d=GCD(a,b).

Then there are such

elements u,v in R, that au+bv=d.

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