# Math Help - Annihilator

1. ## Annihilator

So far i've proved that Ann(a) is a subring (under addition and multiplication), now to prove it's an ideal i've shown;

say a1, a2 elements of R, then a1 - a2 element of R, now i need to show rK, Kr to be an element of R...how do I go about this?

2. Originally Posted by pkr

So far i've proved that Ann(a) is a subring (under addition and multiplication), now to prove it's an ideal i've shown;

say a1, a2 elements of R, then a1 - a2 element of R, now i need to show rK, Kr to be an element of R...how do I go about this?
If $x\in \text{Ann}(a)$ then $ax=0$. But then $r(ax) = 0 \implies a(rx) = 0 \text{ and }a(xr) = 0$. Thus, $xr,rx \in \text{Ann}(a)$.

3. Ah, my version was alot more long winded, thanks very much, sorry I can't add to your reputation, it says i have to give it to others! Many thanks.