3. Are the rings 2Z, 3Z isomorphic. Any isomorphism is linear for addi- tion, so it has the form f (x) = kx. Is there some k which commutes with multiplication? Another approach is to ask if the second ring has a nonzero element y such that y? = 2y.

15. (a) Show that S= {(a,a) | a e Z} is a subring of ZxZ. (Use the definition of addition and multiplication of direct products.) (b) Determine if T = {(a,-a) | ae Z} is a subring of ZxZ. 16. Let R be a commutative ring with unity and let

5. (*) Let R be a ring with 1 in which ab 0 whenever a, b E R are nonzero. Let a ER\ {0}. An element b E R is said to be a left (respectively, right) inverse for a if ba = 1R (respectively, ab = 1R). Show that if a E R has a left inverse b then b is also a right inverse for a (and thus is an inverse for a).

. Let R = Z. Find 5 examples of principal ideals. . Let R = R[x]. Find 5 examples of principal ideals . Consider the ring (P(Z), Δ,∩). What is ⟨{2}⟩?