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6. Let R be a ring and a € R. Prove that (i) {x E R | ax = 0} is a right ideal of R (ii) {Y E R | ya=0} is a left ideal of R (iii) if L is a left ideal of R, then {z E R za = 0 Vae L} is a two-sided ideal of R NB: first show that each set in 6.(i), (ii), (iii) above is a subring T ool of P Prove that

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6 Let R be a ring and aer Prove that (i) {XERlax=} is a night ideal of R. Proof: Let I. {XER lax = of and suppose XEI and YER TO PROVE: a (x) = 0 Then (x-8) EI if a(z.v) = 0 (*: XEI => right side nightidal so, al.) = ax) ( Ring has associative property) (ax=o) oor 3) a (x = 0 -) (x-3) EI I is right ideal of R. ii) {YER Iya=0} is a left ideal of R. Proof: Let J = {YERlya=of. мифы и нет - a = 0 and TeR. Thertyy)a=r (y) 1 Ring Rhas associative property) 1 Idside = 8.0 1 ya = o being YEJ) =) (8y) a = 0 =) & ry EJ J is a left ideal of R.

(ii) Proof If L is a left icual of R, thin & ZER| za = 0 VaEL} is a two-sided ideal of R. het K = {ZER| za=0 VacL} : L is left ideal of R. i VEER, VEL all have saEL ③ left side Now, Let ZEK = za=0 VOEL and XER Then (8.2) a = 8(za) (ZEK = ZER T EL becouse KSR left =80where b=2aEL (za=0 O () => 10.2) = 0 * 8.z EK = K is lift ichal of R. Vael) Next, associativity) (28) a right = z (ra) EL = 2.18a) - 1. TER, GEL - TOEL Left ideal! zek, saeL > Z (wa) = 0 Vale ) c = z8EK k is right ideal of R. Hence K is both sided ideal of R.
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