- #1

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proof

if a=bu, then ar=bur. Since ur in R, call it s. So ar=bs for some s in R. Therefore a times some elements in r is equal to b times a different element in R. Therefore {ra: r in R}={rb: r in R}.

if <a>=<b>, then {ra: r in R}={rb: r in R}. That means ra=r'b for some r,r' in R. However, r'=ru for some unit u in U(R). So ra=rub. Hence, a=ub=bu *commutative ring*.

Therefore <a>=<b> if, and only if, a=bu for some unit u in U(R).

PLEASE HELP I KNOW THIS PROOF HAS SOMETHING WRONG WITH IT.