When the ring, R, is an integral domain. That is, when R has no zero divisors. Consider:
ab=cb => ab-cb=0 => (a-c)b=0
Then because R has no zero divisors we have:
a-c=0 => a=c
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Given that ab = ba and bc = cb We can arrive at abbc = cbba by adding equal quantities to both sides of the equation By the cancellation law you're allowed to drop the bb from both sides of the equation to end up with ac = ca
To prove a ring is commutative, one must show that for any two elements of the ring their product does not depend on the order in which you multiply them. For example, if p and q are any two elements of your ring then p*q must equal q*p in order for the ring to be commutative. Note that not every ring is commutative, in some rings p*q does not equal q*p for arbitrary q and p (for example, the ring of 2x2 matrices).
Ring is to sting as bee is to tree (or see or any other rhyming word).
I have a very old gold colored ring with a clear stone that has fs 4 k inside? what does that mean? what could the stone be? It is not a diamond but the Gold is real and looks like my 14 k ring..
N!/N