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3.10 If U, V are ideals of R, let U + V = {u + v | u ∈ U, v ∈ V }.

Prove that U + V is also an ideal.

3.9 Show that the commutative ring D is an integral domain if

and only if for a, b, c ∈ D with a #= 0 the relation ab = ac implies that

b = c.

3.8 D is an integral domain and D is of ﬁnite characteristic,

prove that the characteristic of D is a prime number.

3.7 If D is an integral domain and if na = 0 for some a #= 0 in

D and some integer n #= 0, prove that D is of ﬁnite characteristic.

3.6 If F is a ﬁeld, prove that its only ideals are (0) and F itself.

3.5 If U is an ideal of R and 1 ∈ U, prove that U = R.

3.7. If D is an integral domain and if na = 0 for some a #= 0 in

D and some integer n #= 0, prove that D is of ﬁnite characteristic.

3.6. If F is a ﬁeld, prove that its only ideals are (0) and F itself.

3.5. If U is an ideal of R and 1 ∈ U, prove that U = R.

3.4. Prove that any ﬁeld is an integral domain.

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