3.3 Ideals and Factor Rings

1.
a. No. 1 img A, AR.
c. Yes. img, img
e. No. 1 img A.
3.
a. (1 + A)(r + A) = (1 · r + A) = r + A and (r + A)(1 + A) = (r · 1 + A) = r + A.
c. If R is commutative then, for all r + A, s + A in R/A :

img

4.
a. mr + ms = m(r + s) img mR, −(mr) = m(− r) img mR; s(mr) = m(sr) img mR; (mr)s = m(rs) img mR. If mr = 0 and mt = 0 then m(r + t) = mr + mt = 0; m(− r) = − mr = 0; m(rs) = (mr)s = 0; and m(sr) = s(mr) = 0 for all s img R.
5.
a. A × B is clearly an additive subgroup and (r, s)(a,

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