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If $R$ is a UFD and $a,b \in R$ are relatively prime and $a|bc$, then $a|c$.

we know if a and b relatively prime (a,b)=1 so $a\bot b$. since $a|bc$ and $a\bot b$ doesn't $a$ has to divide $c$? I dont understand why do we need UFD?

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    or since it is ufd, let's assume $\displaystyle a \;=\; p_1\cdots p_l, \quad b \;=\; q_1\cdots q_m, \quad c \;=\; r_1\cdots r_n$ $\displaystyle p_1\cdots p_l\,=\; q_1\cdots q_m \; r_1\cdots r_n\,k$ , $k\in R$ . and since $a\bot b$ , $q_j$ cant be associate of one of $ p_i$'s . so each one of $ p_i$ must be associate to $q_j$'s and k, then what? :)2012-12-13
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    What do yu mean by $a\bot b$? How is it different from $(a,b)=1$?2012-12-13
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    I meant a cant divide b2012-12-13

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