Anonymous
Anonymous asked in Science & MathematicsMathematics · 10 years ago

left Ideal and ideal of ring R?

1)if a in R, Show that {r inR|ra=0} is a leaft ideal of R?

2)if H is a left ideal of R .show that the set {r inR|rx=0, for each x in H} is an ideal of R

3)if H is an ideal of R .show that [H:R]={r inR|xr in H , for each x in R} is an ideal of R

I Think we just use the conditions for ideal If a,b in { r inR|rx=0, for each x in H} , then so are a−b and ra for all r in R

i hope to help me please show your work thank you

Relevance
• 10 years ago

1) Let L = {r in R| ra = 0}.

a) If u and v are in L then (u+v)a = ua +va =0 +0 =0, so u + v is in L.

b) If r is in R and u is in L then (ru)a = r(ua) = 0, so ru is in L, which shows that L is a left ideal of R.

2) Let L = {r in R| rx = 0 for each x in H}.

a) As in 1) if u and v are in L then (u+v)x = ux +vx = 0 for each x in H, so u + v is in L.

b) If r is in R and u in L then (ru)x = r(ux) = 0 for each x in H, so ru is in L.

c) If r is in R and u in L then (ur)x = u(rx), but since H is a left ideal rx is in H, so u(rx) = 0 for each x in H so ur is in L.

This proves L is an ideal of R.

3)

a) As above if u and v are in [H:R] then x(u + v) =xu + xv = 0 for each x in H, so u + v is in [H:R].

b) If r is in R and u in [H:R] then x(ur) = (xu)r = 0 for each x in H so ur is in [H:R].

c) If r is in R and u in [H:R] then x(ru) = (xr)u, but since H is an ideal then xr is in H, so (xr)u = 0 for each x in H so ru is in [H:R].

This proves [H:R] is an ideal of R.