Please help me with Proof. ::
Let a, b, and c be elements of a commutative ring R with a and b ≠ 0R. If a divides b and b divides c, the prove that a divides c. [This is call the transitive property for division in a ring R.]
(Note that in a commutative ring R where s and t are elements in R, then s divides t if there is an element b such that t = s*b AND s ≠ 0R.)

Question

Please help me with Proof.  ::
Let a, b, and c be elements of a commutative ring R with a and b ≠ 0R. If a divides b and b divides c, the prove that a divides c. [This is call the transitive property for division in a ring R.]
(Note that in a commutative ring R where s and t are elements in R, then s divides t if there is an element b  such that t = s*b AND s ≠ 0R.)

kinggeorge · Answer

Well, we know that $a|b$, so $b=ar$. Then, $b|c$, so $c=bs$. Now you just need to replace the $b$ in $bs$ the $ar$, and it should be relatively straightforward to finish this up.

anonymous · Answer

not ignoring. just trying to figure it out

anonymous · Answer

@KingGeorge   I think i got it!  one sec while i type it

anonymous · Answer

Given a, b, c are elements in a commutative ring R, with  a and b not equal to 0R.  By the transitive property of division, a divides b and b divides c.  By definition of divides, there exists elements s, t such that b = as and c = bt.  Since c = bt, then by substitution, c = (as)t.  Hence c = a(st) by associativity  under multiplication.  Therefore, by definition of divides, a divides c.

kinggeorge · Answer

Looks good to me.

anonymous · Answer

AWESOME!  thank you

kinggeorge · Answer

You're welcome.